Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 963
Force Budget Analysis
of Glacier Flow
Ice Dynamical Studies on Storglaciären, Sweden,
and Ice Flow Investigations of Outlet Glaciers in
Dronning Maud Land, Antarctica
BY
JIM HEDFORS
ACTA UNIVERSITATIS UPSALIENSIS
UPPSALA 2004
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Don’t fight forces, use them.
Buckminster Fuller
To Iréne and Kalle
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals:
I.
Hedfors, J., V. Peyaud, V.A. Pohjola, P. Jansson and R. Pettersson,
2003. Investigating the ratio of basal drag and driving stress in relation
to bedrock topography during a melt season on Storglaciären, Sweden,
using force budget analysis. Annals of Glaciology 37, 263-268.
II. Hedfors, J. and V.A. Pohjola. Seasonal variations in the ice dynamics
on Storglaciären, Sweden. Manuscript.
III. Pohjola, V.A and J. Hedfors, 2003. Studying the effects of strain
heating on glacial flow within outlet glaciers from the Heimefrontfjella
Range, D.M.L., Antarctica. Annals of Glaciology 37, 134-142.
IV. Pohjola, V.A., J. Hedfors and P. Holmlund. Investigating the potential
to determine the upstream accumulation rate, using mass flux
calculations along a cross-section on a small tributary glacier in
Heimefrontfjella, D.M.L., Antarctica. Annals of Glaciology 39. In
press.
V. Hedfors, J., V.A. Pohjola. Ice flux of Plogbreen, a small ice stream in
Dronning Maud Land, Antarctica. Annals of Glaciology 39. In press.
Papers I, III, IV and V are published with permission from the publishers.
My contribution to the papers:
In papers I, II and V, I carried out most of the field work and was
responsible for the data analysis and writing. In paper III and IV, the field
work and writing was carried out by the first author. I contributed to these
papers by jointly running the numerical calculations on which the
discussions and conclusions are based.
Contents
1. Introduction................................................................................................7
2. Aim of the study.......................................................................................11
3. The concept of ice flow............................................................................13
4. Messages from Antarctica........................................................................16
5. Methods and techniques...........................................................................19
5.1. Theory ..............................................................................................20
5.1.1. Glen's law .................................................................................20
5.1.2. Ice temperature .........................................................................22
5.1.3. Force budget .............................................................................24
5.1.4. Ice flux ......................................................................................26
5.1.5. Summary of calculations ..........................................................27
5.2. Limitations .......................................................................................27
5.2.1. Glen's law .................................................................................27
5.2.2. Thermodynamic modeling........................................................28
5.2.3. Force budget .............................................................................28
5.2.4. Errors associated with data collection ......................................29
6. Field sites .................................................................................................31
6.1. Storglaciären, Sweden ......................................................................31
6.2. Bonnevie-Svendsenbreen, Antarctica ..............................................31
6.3. Kibergbreen, Antarctica ...................................................................34
6.4. Plogbreen, Antarctica .......................................................................34
7. Results......................................................................................................36
7.1. Testing the model for spatial elements (Paper I)..............................36
7.2. Testing the model for temporal elements (Paper II).........................38
7.3. Testing a thermodynamical model (Paper III)..................................40
7.4. Mass flux of Bonnevie-Svendsenbreen (Paper IV) ..........................42
7.5. Mass flux of Plogbreen (Paper V)....................................................44
8. Discussion ................................................................................................48
8.1. Force budget tests on Storglaciären..................................................48
8.2. Force budget applications in Antarctica ...........................................49
9. General conclusions .................................................................................54
10. Acknowledgements................................................................................57
11. Summary in Swedish .............................................................................59
12. References..............................................................................................61
Appendix A: Symbols...................................................................................65
Appendix B: Constants .................................................................................66
Appendix C: Force budget ............................................................................67
Appendix D: Flow chart................................................................................70
1. Introduction
Earth scientists often describe the organic and inorganic processes of the
Earth as a web of interactions between five different spheres. These spheres
are commonly classified as i) atmosphere, ii) biosphere, iii) cryosphere, iv)
hydrosphere, iv) lithosphere, and exist in a mutual struggle for equilibrium
governed by natural cycles. Some would also add a sixth sphere, the
anthroposphere, to represent the activity of humans (thus, extracting us from
the biosphere). A change in one sphere has a direct or indirect effect upon
the others, which has become an interesting issue as we, the humans, seem to
have achieved the capability to extensively disturb the major natural cycles
of our environment. That is, if we consider ourselves detached from the
natural system.
At present, there is a debate on whether water stored as ice within the
cryosphere (i.e. ice caps, glaciers, permafrost etc) is being released to the
adjacent spheres and causing unpredictable disturbances in the natural
cycles. For example, the meltwater of a rapidly shrinking glacier within the
cryosphere may cause a slight sea level rise within the hydrosphere. The
properties of the cryosphere is controlled by topography (i.e. the lithosphere)
on long time scales (millions of years) and climate (i.e. the atmosphere), on
short time scales (tens to thousends of years). The changes we observe in the
cryosphere would therefore mainly be an indirect effect of changes
introduced to the atmosphere by antropogenic activity in combination with
natural causes. Thus, the coupling between atmosphere and cryosphere is
obvious and can be used to understand glacier response to climate change.
It is becoming clear that the annual global mean temperature of the last
centuries is rising (Figure 1:1). In addition, the Intergovernmental Panel of
Climate Change (IPCC) predicts a global temperature increase of about 2 ºC
over the next 100 years. In the polar regions, the increase is expected to be
•4 ºC over the same period (Figure 1:2). The reasons for this could be many,
but one of the factors that affect the global annual mean temperature is
probably the increase in anthropogenic release of green-house gases to the
atmosphere. The major contributors, CO2, N2O, CH4 and SO2, all show
trends of strong accumulative concentrations over the last century (Figure
1:3 show trends for CO2 and CH4). In the perspective of increased greenhouse gases and global warming, glaciers in high latitudes (where the effects
7
Figure 1:1 Global temperature trend over the past 140 years*.
Figure 1:2 Predicted global temperature changes over the next 100 years*.
Figure 1:3 Trends of atmospheric concentrations of two green-house gases,
CO2 and CH4*.
*
Published with permission from the Intergovernmental Panel on Climate Change.
8
of global warming is expected to be highest) act as an alarm system
providing early warning signals of global climate change.
About 4 billion people lives within a narrow strip of land at the ocean
edge today. Over 100 million of them are dependent on daily activities
carried out within one meter of the current sea level. Even slight rises in sea
level with subsequent shoreline retreats, have major consequences on many
of our societies. Different future scenarios predicts a sea level rise between
0.1-1.0 m for the next 100 years depending on various melt-rates. The IPCC
2001 compilation, based on tide-gauge data, reports a 20th century rise of
1.0-2.0 mm a-1 with no discernable acceleration (Church and Gregory, 2001),
however Cabanes et al. (2001) show a current rise of 3.2±0.2 mm a-1 (19931998) based on sea level measurements from the TOPEX-POSEIDON radar
altimeter. Thermal expansion accounts for about half of this increase and
land ice melt about 20-25 %. The remaining part comes from changes in
ground and surface water storage associated with consumptive use for
agriculture and industrial production. Thus, the scenario of melting ice caps
and rising sea levels does not describe the future, it takes place today. Ice
caps are melting and the sea level is rising as you read. The question is
where glaciers are melting and where they are growing? How large would a
contribution of melting ice be to the global sea level rise? Do we have to
worry?
At present, the largest contribution of meltwater comes from shrinking
mountain glaciers (Haeberli, 1998) while the ice sheets of Greenland and
Antarctica are still uncertain components. Table 1 presents contributions to
sea level rise from glaciers in various parts of the Earth over the period
1901-1962 (Haeberli, 1998).
Table 1:1 Contributions to sea level rise from meltwater in mountain glaciers.
Region
Alaska-northwestern Canada
Central Asia including the Himalayas
Southern Andes
Arctic islands of North America and Eurasia
Other
contribution (%)
34
16
12
10
28
A shift of the regional balance over the last 30-40 years appear in a report
from Dyurgerov and Meier (1997) where Svalbard and the Arctic islands
have shown a sudden loss (measured between 1961-1993) of ice
corresponding to nearly half of the total contribution in the northern
hemisphere.
These mesurements are based on the melting of most of the small glacial
systems, although including some parts of Greenland. When it comes to the
9
contribution from the entire ice sheets of Greenland and Antarctica, recent
investigations give various answers. These two ice sheets contain enough ice
to raise the global sea level by more than 80 m were they to melt completely.
The continent of Antarctica alone would stand for about 75 %, or 60 m, of
this sea level rise. The amount of snow deposited annually on its surface is
equivalent to 5 to 6 mm of the global sea level (Frezzotti, 2003). Jacobs
(1992) elaborate on the Antarctic ice sheet's response to higher temperatures;
some argue for an increase of the ice volume over the continent as a result of
higher precipitation/accumulation rates from warmer and more moist air
masses; others suggest lower rates of precipitation due to a decrease in
cyclogenesis from more isotropic air masses. Thus, it is too early to say how
the Antarctic ice sheet will behave in a warmer world. Various sources
speculate in, and in some cases confirm, the collapse of large ice sheets as a
direct consequence of higher temperatures (Wingham et al., 1998). The
removal of ice sheets, in turn, enables the large continental ice streams to
accelerate its discharge of ice into the oceans. Melting of large ice sheets
could thus become a major contributor to the present day rise in sea level.
By measuring ice sheet/glacier growth and melt and changes in geometry
and motion, it is possible to draw conclusions about the ice mass balance of
a glacial system. The mass balance between input (precipitation, redistribution of snow by wind etc) and output (sublimation, snow drift,
removal, melt runoff, calving, evaporation etc) tells us whether the ice is
adjusting to new environmental conditions, for example, an increased air
temperature or decreased accumulation rates of snow. There are two basic
approaches to measuring the mass balance of an ice sheet/glacier. The first is
an integrated approach in which changes in mass are measured by remote
sensing techniques via ice surface elevation changes without separatedly
determining the input and output mass fluxes (Krabill et al., 1999). The
second is a component approach in which the imbalance between ice mass
inputs and outputs are individually measured for selected glaciers (Clarke,
1987; Hooke, 1998). Both approaches are not only important for determining
the mass balance, but also for understanding the causes of mass balance
fluctuations. The latter is particularly useful for monitoiring sensitive
glaciers in relation to trends in the global climate.
Despite all available measurements of snow accumulation, ice velocity,
surface and basal melting, and iceberg calving, it is still not known whether
some major ice sheets (including the Antarctic ice sheet) is growing or
thinning (Frezzotti, pers. comm., 2003). It is thus important to define the
glacial processes that control sea level rise, so that future changes can be
predicted and remeidal actions undertaken.
10
2. Aim of the study
The general purpose of this thesis is to contribute to the overall
understanding of ice mass response to climate change. Specifically, it aims
to apply an existing method for estimating the outgoing mass, or ice outflux,
of a given glacial system via studies of ice dynamics. By using established
laws of glacial deformation and force budget calculations for well confined
areas of glacier flow, it is possible to estimate the outflux component
through a glacier cross section, or a gate (Clarke, 1987; Whillans and
Bindschadler, 1988). The outflux is evaluated for temporal changes and/or in
relation to influx in terms of mass balance to provide signals of climate
changes. The key to understanding the outflux lies in the dynamic properties
of the glacier ice at the gate and requires information on ice surface
velocities and ice geometry together with certain climatic parameters such as
temperature and snow accumulation rates. The methodology is tested on a
well known sub-arctic control glacier to motivate further case studies in
polar regions, i.e. Antarctica. The structure of the work needs to recognize
three subordinated aims:
1. collecting and processing of geophysical and remotedly sensed
information of ice movement and geometry;
2. improving on numerical algorithms in order to calculate ice flow via
inversion of the flow law for ice;
3. presenting spatial and temporal information on ice dynamics and mass
fluxes of case studies of polar glaciers.
The first two aims (1 and 2) involve the setup of productive field campaigns
and phases of establishing data management and analysis techniques. Aim 3
is the product of aim 1 and 2 presented in this thesis as Paper I - V.
At present, there are numerous detailed and excellent mass balance
studies of small valley glaciers across the globe where the abundance of data
is the primary factor of knowledge. However, these glaciers would have a
minor impact on the global sea level if they were to melt completely in the
near future. In polar regions, the situation is the opposite: vast areas of huge
ice masses subjected to the highest (expected) warming and with a large
potential to raise the global sea level, but poorly known due to lack of data.
In the light of glacier controlled sea level rise, this work is important since it
11
accounts for ice flux estimations in crucial areas of limited information, i.e.
Dronning Maud Land (DML), Antarctica.
12
3. The concept of ice flow
Glaciers in different parts of the Earth flow with different velocities,
depending on the mechanism of flow and the balance between driving and
resisting forces. Ice flow velocities have been found to range between 0-300
m a-1, but in some extreme cases, as with Jacobshavns Isbrae, Greenland,
velocities of up to 8360 m a-1 have been observed (Lingle et al., 1981;
Echelmeyer and Harrison, 1990). The driving force behind ice- or glacier
flow is gravity, g (acting on every medium with density U), where the total
motion can be partitioned into components of internal ice deformation (creep
and fracture), basal sliding and till deformation. Glen (1955) formulated an
expression that described deformation rate, or strain rate, H , and accounted
for the effects of the gravity-dependent driving stress, W, and the temperature
dependent viscosity, B:
n
§W ·
H ¨ ¸
(3:1),
© B¹
where n is a flow exponent describing the maturity of the ice. The value of n
in Glen’s power flow law has been determined to vary between 1.5 to 4.2
with a mean of about 3 (Paterson, 1994). This exponent makes the strain rate
highly dependent on driving stress since with a value of n of 3, a doubling of
the driving stress produces an eightfold increase in strain rate. The driving
stress, W = UghsinD, is primarily controlled by ice thickness, h, and ice
surface slope, D, causing the driving stress to increase or decrease with
changes in topography. This could be due to an alteration in mass balance;
e.g. an increase in snowfall in the accumulation area will increase the ice
thickness and create a steeper surface gradient, which in turn causes higher
shear stresses. This results in an accelerated ice flow to compensate for the
additional input so that the larger ice volume is now more rapidly discharged
to the ablation area. Conversely, a growing ablation will thin out the ice and
reduce the shear stresses and velocities. The viscosity, B, follows from the
Arrhenius relation:
B
B0
§ Q ·
¨
¸
e © PT ¹
(3:2),
13
where B0 is a constant independent of temperature, T. Thus, higher values in
temperature, T, will reduce the viscosity, B, causing the ice to flow faster. P
is the universal gas constant (8,321 J mol-1 K-1) and Qa is the activation
energy for creep (Qa=78.8 kJ mol-1; Hooke, 1981). Hook (1981) found the
expression for B to be valid for empirical data so that B=B0e(To/T), and with a
modification based on numerical models B0 = 1.928 Pa a1/n and T0 = 3155 K
(Pohjola, 1993; Hansson, 1995). In Glen's experiments, the value of B was
found to be significantly affected by differences in temperature, changing
from 200 kPa a(1/n) at 0 qC to over 400 kPa a(1/n) at -13 qC. Consequently, two
different regimes of ice flow can be distinguished; warm-based glaciers,
which are warm enough to keep temperatures at pressure melting point at the
bed and cold-based glaciers, which are everywhere frozen to their beds.
Fluctuations in the basal water pressure in warm-based glaciers influence the
velocity by altering the resistive forces acting at the bed. Over long time
periods, changes in velocity can also occur in cold-based glaciers as a result
of changes in shear stress- and driving stress conditions.
Warm-based glaciers, common in mid- and low latitudes, generally
maintain high velocities from basal sliding over a lubricating waterfilm,
sometimes with substantial till deformation. Deformation rates are usually
low, but can become pronounced in areas of complex bedrock geometry and
high surface slopes. Storglaciären, Sweden, is one example of a warm-based
valley glacier in a sub-arctic environment that exhibits various basal sliding
rates over time and space as well as complex patterns of deformation in
regions where the bedrock is known to undulate (Hooke et al., 1989). Paper I
and II present an attempt to explain the spatial and temporal variations in the
ice dynamics (surface strain rates, basal conditions etc) for such a glacial
system (i.e. Storglaciären) via force balance calculations. Typical for
Storglaciären are complex patterns of uplift and jerky motions in relation to
the bedrock geometry and the drainage status of the glacier, e.g. the
configuration of englacial and subglacial water pathways (Jansson, 1994).
Indications of plug flow, in which no deformation occurs within the ice, and
even local extrusion flow, in which the basal velocities exceed the surface
velocities, sprung from bore-hole measurements on Storglaciären 1985
(Hooke et al., 1987). This phenomenon was correlated to events of high
basal water pressures followed by temporary peaks in meltwater runoff.
Stress fields implied by a model simulation showed that the glacier receives
its largest driving force from a high-slope zone near the equilibrium line, and
that the largest resistive stresses are developed from lateral drag (Hansson,
1995). In the same study, Hansson found that the lateral drag is enhanced on
this glacier because it is frozen to its sidewalls and from turning of the main
flow. Thus, the character of ice flow within warm based glaciers requires the
consideration of seasonality and the availability of meltwater input and its
distribution within the glacier. The character of the bedrock topography and
14
the coupling of longitudinal and lateral forces over geometrically different
areas are also factors of great importance.
Cold-based glaciers, common in high latitudes, can flow only by internal
deformation of the ice and the upper part of the bed, because sliding is small
or negligible at subfreezing temperatures (Boulton, 1979; Echelmeyer and
Wang, 1987). The inverse relationship between ice-creep rates and
temperature (discussed further with Glen's law in chapter 5) means that only
low flow velocities are possible for cold-based glaciers. One good example
of this is the Meserve glacier in Dry Valley, Antarctica, which show a
maximum of only 2 m a-1 (Chinn, 1988). Generally speaking, glaciers in cold
regions have small velocities, but even so, these are sufficient to discharge
the annual ice accumulation. However, where velocities are high, for
instance, where ice from large accumulation areas are funneled through
narrow valleys, the ice thickness will increase and raise the basal
temperature via processes of frictional heating. This process is investigated
in Paper III via thermodynamical modeling and studies of surface velocities
of an Antarctic outlet glacier. It seems like this strain-heating effect can be
sufficient to raise the temperature to the pressure melting point at the bed,
and in some extreme cases, basal sliding takes place to compensate for the
annual accumulation. This means that glaciers in very cold environments
may flow by the mechanism of basal sliding and till deformation, and not
solely by internal deformation.
So, how can we connect our knowledge of ice dynamics to questions of
climate change? This is done by the concept of ice mass balance or balance
velocities. In a balanced glacial system, the flow through a cross-sectional
gate, ‡out, should balance the incoming mass upstream, ‡in (Clarke, 1987;
Hooke, 1998). Due to a temporal change in the mass balance, or in the ice
dynamical situation, this balance may be disturbed. An unbalanced system
indicates a change within the cryosphere, a change that is a response to new
climatic conditions. A negative balance suggests that the glacier is adjusting
to a state of lower accumulation rates or higher ablation rates, and vice versa.
In order to determine ‡in, information on accumulation rates and
accumulation area is required. Accumulation rates can be determined by
direct measurements of precipitation or by the study of ice core records. The
accumulation area can be outlined knowing the elevation conditions at the
site. Here, digital maps and satellite imagery are widely used to reveal
parameters such as slope and aspect of the ice for assessment of ice divide
and other factors of importance to the ice flow. ‡out, can be determined by
studying melt water runoff, evaporation, wind drift, calving of icebergs, but
also through mathematical modeling of ice flow through cross sectional
gates (e.g. Høydal, 1996), of which the latter is the focus of Paper IV and V.
15
4. Messages from Antarctica
The last decade of ice sheet research has identified unexpected changes in
the major ice sheets suggesting a much more dynamic situation than
previously thought. The new concept for Antarctica as being configured
much like a river network, where a web of ice streams drains the entire
interior (Figure 4:1), explains some of the changes and why it is sensitive to
change. Bamber et al. (1992) found that these ice streams cover about 5-10
% of the Antarctic surface, while actually draining 90-95 % of its total ice
mass. The idea of a chunk-like slow evolving ice sheet has thus radically
changed to a view of the ice sheet as driven by feedback mechanisms within
numerous dynamically different key areas, each sensitive in its own way to
both local and global changes in the environment.
DML
Figure 4:1 Location of 33 Antarctic ice streams and outlet glaciers overlaid on a
map of calculated ice sheet balance velocity. Catchment basin boundaries are black,
grounding lines are red, and ice shelves are gray. Reprinted (abstracted/excerpted)
with permision from Rignot, E. and R. Thomas, 2002. Mass Balance of Polar Ice
Sheets. Science 297: 1502-1506. Copyright 2002 AAAS.
16
Recent results indicate differences in the ice dynamics between different
geographical locations, in which the West Antarctic Ice Sheet (WAIS)
appear to loose ice to the ocean and the East Antarctica is near equilibrium
or sligtly imbalanced by a positive flux (from discussions on the 7th
International Symposium on Antarctic Glaciology (ISAG7), Milan, Italy,
2003).
Major changes in the ice dynamics such as changes in ice stream velocity
and direction, glacier retreat, ice shelf retreat and thinning, grounding line
retreat etc, are now occuring in West Antarctica (Joughin and Tulaczyk,
2002), in the Ross sea (Jacobel et al., 2000) and Amundsen Sea sector
(Rignot and Thomas, 2002; Bindschadler, 2002). In total, the glaciers of the
Amundsen Sea sector alone are calculated to have lost over 150 km3 of ice to
the ocean in the past 9 years, which is equivalent to 0.43 mm a-1 of eustatic
sea level rise (Shepherd et al., 2001). Zwally (2002) reports a similar trend
for the same area, although coming forward with a smaller average ice
thinning rate and a corresponding sea level rise of 0.2 mm a-1. Put in context,
the entire WAIS is not in equilibrium with present environmental conditions,
but has been thinning for the last 10 000 years due to an adjustment from the
last glacial maximum (Ackert, 2003; Stone et al., 2003).
Significant changes have also been detected to occur in the Antarctic
Peninsula from studying accelerated flow and glacier surge (De Angelis and
Skvarca, 2003) and ice shelf collapse (Scambos et al., 2000). The changes
are believed to relate to an increase in surface air temperature over the
Antarctic Peninsula of 2.5 qC in the past 50 years (Vaughan and Doake,
1996). This is an order of magnitude greater than the mean global warming
of +0.6±0.2 qC during the 20th century (Figure 1:1).
New discoveries of enhanced basal melting rates experienced by large
outlet glaciers near their grounding line (Frezzotti et al., 2000) seem to be a
controlling mechanism of ice sheet collapse (Huybrechts and De Wolde,
1999). A feedback to accelerated flow of glaciers feeding ice shelves is
achieved as the collapse of the ice shelf eliminates the back-pressure felt by
upstream ice streams (De Angelis and Skvarca, 2003). In addition, ice shelfs
collapse simply because of rising temperatures in the surrounding ocean
(Wingham et al., 1998). Although the collapse of floating ice shelves has no
effect on sea level, it can have a profound influence on ocean circulation,
climate, and the dynamics of tributary glaciers. As Frezzotti puts it: "The
significant sensivity of glacier fluxes (transfer of mass in- and output of
glacial systems) to ice-shelf back pressure caused by ice shelf retreat calls
for reconsideration of importance of the latter process in stability of marinebased ice sheets during atmospheric warming and/or ocean warming" (cited
from Frezotti, ISAG7, Milano, 2003).
Some of the glaciers in East Antarctica show a slight negative imbalance
or states of near equilibrium (Pattyn and Derauw, 2002) while the area as a
17
whole is associated with positive fluxes, although the sign of the imbalance
is rather uncertain (Rignot et al., 2002; Rignot and Thomas, 2002). One of
the major unexpected discoveries regarding atmosphere-cryosphere
interaction was made in the remotest part of East Antarctica when
megadunes and wide glazed surfaces were studied. It showed that surface
wind and subtle variations in surface slope in the wind direction have
considerable impact on the spatial distribution of snow over both short and
very long distances (Frezzotti et al., 2002). This may have an impact for the
interpreteation of mass balances when using ice core records as proxy for
accumulation rates over large areas. A general trend of increased
accumulation rates has been observed in the interior of Antarctica,
specifically at the South Pole (Mosley-Thompson et al., 1999), and in
Wilkes Land (Morgan et al., 1991). Studies in DML show large spatial
variability in accumulation/ablation patterns (Oerter et al., 2000; Isaksson et
al., 1999) but no significant trends can be seen in the overall mass balance
for this region. Nevertheless, the spatial variations in temperature, near
surface firn density and accumulation suggest that katabatic winds affect this
region, in which mass loss by sublimation is small and erosion of snow by
wind has a potentially large impact on the surface mass balance (Van den
Broeke et al., 1999). Ice dynamical investigations on three large ice streams,
Veststraumen (Holmlund et al., 2003), Jutulstraumen (Høydal, 1996), and
Shirase Glacier (Pattyn and Derauw, 2002) in DML have shown fluxes of
outgoing mass of 12, 13.3 and 13.8 km3 a-1 respectively. These studies
indicate a system in balance and no significant trends of increased or
decreased discharge. However, large glacial systems such as these involve
handling of large uncertainties thus making it difficult to establish reliable
mass balances. In this context, additional ice stream balance flux studies in
DML would contribute to the work above. This thesis targets outlet glaciers
in Western Maudheimvidda, DML, 20-30 times smaller than, for example,
Veststraumen, to investigate both input and output fluxes in relation to
climate change. The potential contribution to sea-level change from these
glaciers is thus smaller, but possible trends revealed are important, not only
for the understanding of regional climate induced changes, but also for the
understanding of rapid changes in ice dynamics of the entire Antarctica.
18
5. Methods and techniques
The theoretical background to this work dates back to the power flow law for
ice (Glen, 1952, 1955, 1958) in combination with a more recent study on
force balances presented by Van der Veen and Whillans (1989). In general,
the method finds the glacier outflux by calculating ice velocity with depth at
a cross sectional gate using Glen's expression of strain rate as a function of
shear stress and ice temperature. The shear stress is given by force balance
calculation techniques and the ice temperature can be assumed a constant
value (where found fit) or modeled using a thermodynamical model.
This approach requires input data in the form of ice surface velocities, ice
geometry, surface temperature and accumulation rates of snow. Glacier
velocities are captured using DGPS (Differential Global Positioning System)
surveying of stake nets planted on the glacier surface (Figure 5:1a) and the
geometry, i.e. ice depth, is obtained by GPR (Ground Penetrating Radar)
profiling along numerous transects (Figure 5:1b). Temperatures and
accumulation rates used here have been provided by previous studies of
meteorological data and ice core records. The data has been complemented
b
a
Figure 5:1 a) DGPS surveying of stake nets on the outlet glacier Kibergbreen, which
drains Amundsenisen through the Heimefrontfjella escarpment, Antarctica. b) GPR
profiling on the ice stream Plogbreen, DML, Antarctica. Photo: Josef Källgården (a)
and Jim Hedfors (b).
19
by studies of AVHRR-based photoclinometry derived digital elevation
models, and surface features from RADARSAT-1 synthetic aperture radar
(SAR), Landsat TM and feature tracking from sequential SPOT 10 m
panchromatic data.
For mass balance purposes, the outflux is compared with the influx
estimated by integrating accumulation rates of snow over the total catchment
area upstream the gate. The following is a presentation of the theory behind
the calculations and its limitations.
5.1. Theory
The theory behind the solution involves three phases: i) establishing the
dynamical situation, i.e. distribution of driving and resistive stresses at the
bed for a section of the glacier/ice stream; ii) estimating ice temperature
profiles; and iii) calculating velocity with depth in order to determine
mass/volume flux through a cross-sectional gate. The first and second phase
(i and ii) provide the input for the third (iii) phase via iterative algorithms of
force budget calculations and thermodynamic modeling. The scheme
assumes horizontal glacier flow along the x-axis direction. The y-axis
represents the horizontal direction perpendicular to the major flow direction,
or transverse glacier flow, and the z-axis represents vertical components.
This simplifies the description below by mainly considering the components
needed in strive for mass fluxes through a cross-sectional gate. Expressions
of full geometric configurations can be found in the force budget theory by
Van der Veen and Whillans (1989). Presented below are the major numerical
steps in a somewhat reversed order motivated by the importance of the
fundamental laws followed. Symbols and constants introduced here are also
listed in Appendix A and B respectively.
5.1.1. Glen's law
The variation of velocity with depth is easily calculated by assuming simple
shear, W = Wxz, meaning Wxz is the only non-zero component. This is described
as laminar flow since the z-component of velocity is zero and the flow lines
are paralell to the surface. At surface slope D, ice surface elevation s, and ice
bed elevation b, the shear stress at ice depth (s-b) is:
Wxz = Ug(s-b)sin D
(5:1),
where U is ice density and g is gravitational acceleration. Deformation, or
strain rates, H , is the change in velocity, U, along any given axis. With
H =½wu/wz, Glen's flow law gives:
20
1 wu
2 wz
§ W xz ·
¨
¸
© B ¹
n
(5:2).
Integration of Equation 5:2 with the value of Wxz given by Equation 5:1,
gives:
n
n 1 ·
§ Ug sin D · §¨ h
¸
U xs U x (h) 2¨
¸ ¨
(5:3),
B ¹ © n 1 ¸¹
©
where Uxs is surface velocity and Ux(h) is the velocity at depth h. A
development of this expression can account for velocities in extending and
compressing flow allowing longitudinal strain rates to modify the velocity
distribution. If the ice thickness varies only slowly along the main flow
direction, it can be assumed that:
Hxz
wU x
wz
§ 1 ·
2¨ n ¸W e ( n 1)W xz
©B ¹
(5:4),
where Ux is ice velocity along the main flow direction (Paterson, 1994). A
thorough discussion on bedrock bumps and hollows in relation to Equation
5:4 is presented in Reeh (1988). Equation 5:4 introduces the effective shear
stress, We, which is a stress deviator that increases the strain rate at all depths
from its value of simple shear. This also increases the difference in velocity
between the surface and the bed. Figure 5:2 illustrates the use of Equation
5:4 as strain rates, H xz, is calculated for half a cross section of Storglaciären
with given basal drag.
1340
Glacier surface
1320
0.005
1
0. 0
1300
Elevation m.a.s.l.
0.
5
01
1280
0. 0
1260
2
25
0. 0
1240
0. 03
0.035
Bedrock
0.04
1220
45
0. 0
1200
7.5345
7.5346 7.5346 7.5347 7.5347
Cross section coordinate [m]
7.5347
7.5348
6
x 10
Figure 5:2 An example of strain ratets ( H xz) calculated for half a cross section of
Storglaciären using Equation 5:4.
21
Numerous observations on glacier behavior compare very well with
values obtained with Glen's general formula (Equation 3:1). First, it is noted
that the strain rates (i.e. degree of deformation) are large at the bed, where
both stress and ice temperature reach maxima (in the case of a cold-based
ice). Second, it explains why a warm ice moves faster than a cold ice. Third,
it can be shown how the glacier controls its own loss by a feedback
mechanism. Since an increase in ice thickness increases the basal shear
stress, an acceleration of the ice flow is attained to eventually reduce the ice
thickness. An increase in ice thickness and slope gradient also contributes
towards a warmer ice base, and consequently a faster flow, due to strain
heating (if the heat gain overpowers the cooling due to advection from the
surface) so that a higher mass accumulation (input) will be compensated for
by an accelerated ice flow (output) (Pohjola and Hedfors, 1997). Also,
stresses within the ice can not always be compensated for by a change in
deformation, resulting in movement by ice fractioning. Generally, this
occurs at marginal zones where large gradients in frictional resistance
generate high shear stresses, which give rise to large crevasses.
Below are descriptions of the methodology in obtaining, first, the ice
temperature needed for the viscosity calculations, second, the force balance
that provides information on the stress components, and finally the mass flux
calculation through a gate.
5.1.2. Ice temperature
Ice viscosity is known to decrease as temperature increases, leading to
increased deformation rates in accordance with Glen’s flow law (Paterson,
1994). Hence, much of the deformation takes place in the bottom layers
where temperature is raised due to geothermal heat and heat production from
internal heat sources. The general equation for heat transfer is a partial
differential equation, where heat is conducted and advected through the ice
system between the atmospheric and the lithospheric boundaries in all three
dimensions (Budd, 1965; Paterson, 1994):
Uc
wT
wt
§ wT
wT ·¸
’ K’T Uc ¨¨ u
v
wy ¸¹
© wx
· wT
§ dK
Q
¨¨
Ucw ¸¸
¹ wz
© dz
(5:5),
where T is temperature, t is time, c is specific heat capacity, K is thermal
conductivity, U is ice density, u, v, w are the horizontal (along-flow), lateral
(across-flow) and vertical ice velocities, and Q is heat generated from
internal and external heat sources. The parameters c and K are functions of
22
temperature and their empirical relations are given in Paterson (1994). The
first term on the right side in Equation 5:5 is the conduction differential, and
the three following terms are advection terms. Some simplifying
assumptions enables the equation to be treated more easily numerically.
These assumptions are as follows:
‰
‰
‰
‰
The largest temperature gradient is the vertical one, why the
conduction is most effective along the vertical axis. This eliminates
the y och x dimensions.
If we use a model where an ice-column is moving along a flowline at
the centerline, we can exclude the transversal vector. According to the
definition, there is no transverse movement at the centerline. This
eliminates all but the vertical advection term in the heat equation.
The heat production is limited to internal frictional heat, or
deformation heat: Q=Qd. This assumption neglects the latent heat
flux, which implies that the model is only valid for temperatures
below the melting point.
Boundary values at top and bottom are fixed to constant air
temperature and constant geothermal heat exchange.
Thus, only the vertical advection term in the model is used. With the
assumption that the vertical temperature gradient is larger than that in other
directions, (which implies that conduction is only important along the
vertical axis) the heat equation becomes:
Uc
wT
wt
2
§
·
¨ K w T §¨ dK Ucw ·¸ wT Q ¸
¨ wz 2 © dz
¸
¹ wz
©
¹
(5:6),
(Paterson, 1994). The vertical advection, w is assumed to decrease linearly
with depth to a zero value at the bed.
Model input to Equation 5:6 consists of ice surface temperatures (T),
accumulation rate ( a ), ice depth (h), and surface slope (Į). The vertical
velocity, w, is assumed to decrease linearly with depth to a value of zero at
bed. The geothermal heat flux is set to a constant value that represents the
geographic location. Some strictly experimental temperature profiles
illustrate the model output for a few input arguments in Figure 5:3.
23
Temperature profiles for accumulation rate 0.1 m w.e.a-1, surface temperature –20 °C.
Temperature profiles for: Accumulation rate 0.1 m/a, Surface temperature -20oC
0
100
Ice depth [m]
200
300
400
Time=100
Depth=400
Slope=1
Time=200
Depth=400
Slope=2.5
Time=300
Depth=400
Slope=2.5
500
600
Time=100
Depth=600
Slope=1
700
-20
-19
Time=200
Depth=600
Slope=1
-18
Time=200
Depth=600
Slope=2.5
Time=300
Depth=600
Slope=1
-17
-16
-15
Time=300
Depth=600
Slope=2.5
-14
-13
Temperature [ oC]
Figure 5:3 An experimental run of the strain heat model showing calculated
temperature profiles for various input parameters. It is, for example, clear that for
the same time period, a glacier with a 2.5q slope gradient and 400 m thick ice
produces more heat than a 600 m thick glacier with a 1q slope gradient.
5.1.3. Force budget
Van der Veen and Whillans (1989) present a method of calculating stresses
and strain rates at depth taking measurements of surface deformation as
input. The method is: “…applicable to all glaciers: valley and outlet glaciers,
ice shelves and ice streams, as well as inland ice.” (cited from Van der Veen
and Whillans, 1989). The balance, here considering both longitudinal and
transverse directions, is given by:
h
h
w
w
Rxx wz ³ Rxy wz W dx W bx
³
wx b
wy b
h
0
(5:7a, b).
h
w
w
Rxy wz ³ R yy wz W dy W by
³
wy b
wx b
0
These relationships constitute the basic equations needed for calculating
basal drag from driving stress and resistive stress acting on vertical surfaces.
The first term, w/wx³Rxjwz (j=x, y), represents the contrast in resistive forces
acting on the ice block's x-faces and the second term, w/wy³Riywz (i=x, y),
describes the imbalance in resistive forces acting on the ice blocks y-faces.
The third term, Wdi (i=x, y), is the total driving stress in the x- and y-direction
24
and the last term, Wbi (i=x, y), is then the friction at the base of the glacier.
The derivation of Equation 5:7a, b, is found in Appendix C.
The continuation focuses on the situation where the glacier ice flows
parallell to the x-axis direction. By re-arranging the terms in Equation 5:7a
and substituting the integrals by total depth, h, also taking resistive stresses
constant with depth and neglecting vertical resistive stresses, we can write
the so-called isothermal block-flow model for the horizontal direction, x, as:
W dx W bx w
hR xx w hR xy
wx
wy
(5:8),
where the driving stress, Wdx, is balanced by basal drag, Wbx, and resistive
stresses, Rxx and Rxy, for ice depth h. The purpose is to solve for basal drag
(Wbx) which in turn provides the means for calculating the shear stress (We and
Wxz) used in Equation 5:4. The driving stress in the longitudinal direction is
calculated from the glacier geometry through the familiar expression,
Wdx = Ughwz/wx
(5:9),
where wz/wx is the surface slope. Measured surface velocities, Ux and Uy, are
used to determine the strain rates ( H xz and H yz are assumed to be negligible
at the glacier surface):
Hxx
wUx
wx
H yy
wUy
wy
(5:11),
Hzz
ª wUx wUy º
«
wy »¼
¬ wx
(5:12),
Hxy
1 ª wUx wUy º
wx »¼
2 «¬ wy
(5:13),
He
(5:10),
1 2
H xx H 2 yy H 2 zz H 2 xy H 2 xz H 2 yz
2
>
@
(5:14),
from which the resistive stresses at the surface are calculated via deviatoric
stresses, ı' (Rzz=0 due to neglible bridging effects, c.f. Appendix C) as:
R xx
2V ' xx V ' yy
2 BHe1 / n 1H xx BHe1 / n 1H yy
and
25
(5:15),
R xy
V ' xy
BHe1 / n 1H xy
(5:16),
using an assumed average block-value of B corresponding to values
provided by the thermodynamical model. Remembering the force model
taking surface measurements as input, it is necessary to work with deviatoric
stresses in order to calculate resistive stresses. The resistive stress can be
expressed in terms of deviatoric stress under the condition that the normal
deviatoric stress must sum to zero. Deviatoric stresses are full stresses with
the mean pressure removed, as called for by the constitutive relation. For a
further description of the origin of deviatoric stresses, c.f. Van der Veen and
Whillans (1989).
The effective deviatoric stress, We, is described analogous to effective
strain rate as:
We
1 2
V ' xx V ' 2 yy V ' 2 zz V ' 2 xy V ' 2 xz V ' 2 yz
2
>
@
(5:17),
BH1 / n1Hij
i, j = x, y, z
(5:18).
where
V ',ij
Since the calculated resistive stresses act on the body in all planes except at
the base, the force budget must be balanced by a basal shear stress
component, basal drag (Wbx), which now can be calculated as the only
unknown in Equation 5:8. Equation 5:8 is simplified assuming Wxx, Wyy, Wzz and
Wxy are constant with depth, which gives that the depth integrated values
given from the force budget can be used. The vertical stress component, Wxz,
is set to zero at the glacier surface and increases linearly to a value equal to
the basal drag at the bed.
5.1.4. Ice flux
Horizontal ice velocity with depth, Ux, is calculated at the cross-section
using Equation 5:4 with the stress information obtained from the force
balance model above:
U x( h 1)
U xh H xzh ww
wx
(5:19),
where h is level below glacier surface and w is found from the vertical
velocity distribution calculated from the vertical strain rates ( H zz); w is
assumed to be zero at the bed. The ice flux is determined by integrating the
26
calculated horizontal velocity in the longitudinal direction (x-plane) over ice
depth and length of chosen cross section in the transverse direction (y-plane):
s Ac
I out
(5:20),
³ ³ U x dydz
bA
where s-b is total ice thickness and A-A' represents the length of the section
in the horizontal y-plane.
5.1.5. Summary of calculations
As a summary, my studies are based on Equation 5:4, with its limitations
further discussed below. The value of B is calculated as B = B0 e(T0/T); B0 =
1.928 Pa a(1/n) and T0 = 3155 K, where the ice temperature, T, is assumed a
constant value, or in the case of polythermal glaciers, is calculated through
thermodynamical modeling. The stress components, We and Wxz, depend on the
force balance for the gate at the local site and is given by force budget
calculations following Van der Veen and Whillans (1989). Once the strain
rates with depth are known, ‡out is given by integrating velocity over cross
sectional length (y-direction). ‡in can then be compared with ‡out to test for
steady state conditions. A flow chart in Appendix D illustrates the major
elements required in terms of scripting procedures.
5.2. Limitations
5.2.1. Glen's law
It has been found that the flow exponent, n, increases with stress for stresses
greater than about 500 kPa, however, shear stresses in glaciers seldom reach
this value. At stresses below about 100 kPa, n decreases to a value near 1
(Mellor, 1967). Most of the laboratory tests or deformation measurements
(temperate glaciers) provides the value of n=3. This is true in the case of
dynamic recrystallization, but in ice sheets or ice caps, where the
temperature is lower, n should be taken to be between 2 and 3. Paterson
(1994) suggests the best value to use is 3, even in ice sheet modeling
(discussed further below). If assuming that the viscosity is constant, it is also
assumed that the effects of temperature, pressure, water, impurities, density,
and crystal size and orientation are constant as well. This is not true, since all
the mentioned parameters above, in reality, contribute to making the
viscosity highly variable. For a further description of heat flux and viscosity
27
variability c.f. Hooke (1981) and Clarke et al. (1977). Then again, modeling
requires the utilization of limits and simplifications.
5.2.2. Thermodynamic modeling
A major drawback in the calculation technique rests in the assumption and
use of constant or modeled temperature. Along with this assumption, also the
viscosity of ice is given a constant or modeled value throughout the ice,
approximating natural effects such as inhomogeneous crystal size and
impurities.
The use of the temperature, T=0°C, is motivated when dealing with
isothermal glaciers as in Paper I and II. In large outlet glaciers in polar
regions, the ice temperature becomes important as it varies with depth. In an
experiment to model ice temperature with depth (Paper III), we show that a
large portion of energy contributing to higher temperatures, comes from
strain heating - or internal friction due to deformation. The developed
thermodynamic model is iterated until the temperature profiles produce
surface velocities that match measured values. The uncertainties introduced
hereby are thus a combination of errors from measured surface velocities and
the assumptions discussed in section 5.1.2.
5.2.3. Force budget
The force budget model used is simple to carry out but has drawbacks. The
viscous terms in the balance calculation (the last two terms in Equation 5:8)
tend to be overestimated if surface temperatures are applied to the entire
glacier thickness. Hence, the inferred value of basal drag is a limiting value.
The other extreme is found by setting the basal drag equal to the driving
stress (B = 0). The actual basal drag should fall between these two extremes
and this can be accounted for to some extent by applying the lower quartile
of a modeled temperature profile to the isothermal block-flow scheme.
The force balance calculation assumes that the net force acting on any
section of a glacier is zero. This is only true when inertial effects are
negligible (Van der Veen and Whillans, 1989), which is assumed in this
model. Also, it was assumed that ice is an incompressible medium, which is
not perfectly true, since ice is subject to minor compression due to voids.
The bridging effect is briefly discussed in Appendix C to motivate the
exclusion of resistive stresses in the z-direction (Rzz). These vertical stresses
should be included in a more complex model taking into account the vertical
stresses generated by overlaying ice. It has, however, been shown that the
magnitude of this z-parameter is of such small value that it plays a negligible
role in the overall stress situation (Van der Veen and Whillans, 1989).
28
The value of the ice flow exponent, n, has a dramatic influence on the
deformation rate, in some cases causing basal velocities to double with a
change of n from 2 to 4. Borehole studies of Storglaciären were used to
reveal basal sliding rates that guided a tuning of the force budget model to
produce reasonable deformation rates. For the large outlet glaciers studied in
Paper IV and V, we tested a variation of n=3±16 % that produced an error of
±2.1 % in the calculated outflux.
Minor errors in the calculations are incorporated by the use of empirically
determined physical constants and assumptions of boundary conditions as
well as in assumed linear relationships. Ice density is assigned a constant
value for the calculation of driving stress. In reality ice density generally
increases with depth. Not enough information on ice density was available to
make reliable density profile models, therfore a constant value of 900 kg m-3
has been used. Constant ice geometry through time has been assumed,
neglecting the fact that surface slope angle as well as ice depth changes with
time. No practical solution could supply information different from the one
used in the model. Various boundary conditions for the algorithm allow the
model to perform numerically correct, but put constraints on the results
regarding the true physics. Linear relationships are used for stress
components, increasing from 0 at the glacier surface to equal the basal drag
at the bed. Here, the integration steps of vertical stress components also
affect the results, especially for glaciers of great thickness. Mathematically,
these kinds of limitations can be avoided to some extent by using more
sophisticated mathematics.
5.2.4. Errors associated with data collection
The errors introduced by DGPS measurements involve the instruments
accuracy, the tendency for stakes to lean or sway at the moment of
measuring, and the amount of satellites that the coordinate solutions are
based upon. However, the long timeperiods over which the stakes were
measured reduce the instrumental and operational errors drastically for the
glacier surface velocity data sets. Maximum measurement errors have been
simulated by randomly adding/subtracting values within the error range to
each stake position in the DGPS campaigns. Simulations of errors, or
possible stake positions, produce almost identical basal drag patterns across
the investigated glaciers. Differences in values of 10-20 % are found to
cover most parts of the investigated areas, while some larger anomalies
appear in regions of low interest. Even the seasonal changes observed in the
calculations exceed this error significantly. This fact confirms the reliability
of the input data and proves the stability of the analysis method.
Considering the severe ionospheric conditions over Antarctica in
combination with single baseline measurements, the error in achieved
29
positions are probably larger than specified by the GPS manufacturer (Luca
Vittuari, pers. comm., 2003). By using only the relative movement of the
stakes, we apply similar errors to all measurements when calculating
velocity vectors. This way, the data turn out to be very consistent even for
very short measurement intervals.
The ice depth derived from GPR surveys was assigned a large error (r10
%) due to the complexity of interpreting radio waves. The ice depth
information is most reliable at the glacier centerline while moving towards
the margins, the risk of disturbed reflections reduce the quality of the data.
On some sites, the possibility to independently interpret the same transect
twice provided a good tool to estimate the error in the interpretation phase.
This gave errors within r5 %, but additional errors reside in the
mathematics. Refraction effects were neglected and a constant value for
dielectric permittivity of ice was assumed. This has implications on the
result. However, since our measurements fell within r10 % of values
obtained from previous radar sounding campaigns, a r10 % error seems
reasonable.
Errors in incoming flux, ‡in, are large due to difficulties in outlining ice
divides for accumulation areas in polar glacial systems (Price and Whillans,
1998). By using satellite imagery in combination with AVHRR-based
photoclinometry derived digital elevation models to estimate accumulation
area and borehole records of accumulation rates, we found errors in ‡in of
about r23 %. For comparison, Fricker et al. (2000) found an uncertainty of
r20 % in the incoming mass in their precipitation model as they compared
measured and modeled balance fluxes for Amery Ice shelf, East Antarctica.
30
6. Field sites
The sites investigated within my project span from the polar regions of
Lomonosovfonna, Svalbard and DML, Antarctica to sub-arctic Lapland,
Sweden and sub-antarctic Tierra del Fuego, Chile. Relevant to this thesis are
Storglaciären, Sweden (Figure 6:1) and two outlet glaciers, Kibergbreen and
Bonnevie-Svendsenbreen (B-S-breen) at Heimfrontfjella and the ice stream
Plogbreen near Vestfjella, DML, Antarctica (Figure 6:2). A set of
photographs show the field sites in Figure 6:3.
6.1. Storglaciären, Sweden
Storglaciären (67q55'N, 18q35'E) was used as training ground for the
development of field work procedures and equipment testing as well as for
evaluation of data and modeling results. Storglaciären is the largest
(swedish: stor=large) of four valley glaciers scouring the east mountainsides
of the Kebnekaise massif. It covers a surface area of approximately 3 km2
with a total length of 3.2 km from its head at 1730 m a.s.l. to the terminus at
1120 m a.s.l. The average glacier thickness along the flowline is 195 m, with
a maximum depth of 250 m at one of four overdeepenings characterizing the
bed topography. The bedrock of Storglaciären consists of folded gneisses,
amphibolites, and diabase dykes and provides the geological control for the
undulating topography (Andreasson and Gee, 1989). Ice temperatures are
generally 0 qC although a shallow cold surface layer has been observed
(Pettersson, pers. comm., 2003). Ice velocities on the lower section range
between 15-25 m a-1 at the centerline. Additional descriptions of the physical
setting of Storglaciären can be found in Jansson (1994) and Pohjola (1993).
The Storglaciären case study provided valuable information on spatial and
temporal variations in the ice dynamics (Paper I and II) which founded the
base for further mass flux studies on larger outlet glaciers in polar regions.
6.2. Bonnevie-Svendsenbreen, Antarctica
Bonnevie-Svendsenbreen (B-S-breen) (74q45'S, 11q30'W), a 7 km long and
2.5 km wide tributary glacier to the outlet glacier Kibergbreen, flows
31
Storglaciären
69oN
68oN
Bedrock contours
Glacier surface contours
Moraine
Stream
67oN
66oN
65oN
64oN
Kebnekaise north
summit
15oE
18oE
o
21 E
o
24 E
Nordjokk
Sydjokk
N
0
500
Meters
Figure 6:1 Overview of Storglaciären field site
10
0
km
Ritcherflya
Sivorgfjella
Plateu
Figure 6:2 Overview of the field sites in DML, Antarctica
32
a
b
c
d
Figure 6:3 Photographs showing the filed sites; a) Storglaciären seen from the east
with the Kebnekaise south and north summit in the background; b) BonnevieSvendsenbreen draining the Sivorgfjella plateau (upper right) through the
Heimefronfjella escarpment towards Ritcherflya ice field (far left) with
Schönsbergskarvet in the background; c) Kibergbreen, looking upstream) with
Amundsenisen (far left) and Sumnerkammen in the background (far right); d)
Plogbreen, as seen from the Nordenskjöldbasen nunatak, flowing from left to right
with the Plogen mountain massif in the background. Photo: Jim Hedfors.
through the Heimefrontfjella escarpment, DML, Antarctica. The
Heimefrontfjella escarpment is a part of a rift-system from the break-up of
Gondwana, and is sub-divided by outlet glaciers and ice streams into four
separate ranges; Tottanfjella, Sivorgfjella, XU-fjella and Milorgfjella.
Behind Heimefrontfjella the Amundsenisen ridge connects to the Valkyrie
Dome further inland. Bonnevie-Svendsenbreen feeds from a small, local
dome on the Sivorgfjella plateau (~2600 m a.s.l.) and falls down the
escarpment westwards where it joins Kibergbreen at ca 1500 m a.s.l. The ice
depth is about 500 m and surface velocities ~50 m a-1. The case study of
Bonnevie-Svendsenbreen had two tasks. First, to establish a simple, yet
robust, thermodynamical model that could be used for estimating ice
temperature with depth (Paper III), and second, to investigate the potential to
determine upstream accumulation rates via outflux calculations (Paper IV).
33
6.3. Kibergbreen, Antarctica
Kibergbreen (74q50'S 11q42'W), 5-7 km wide and 30-35 km long, flows
through the Heimfrontfjella escarpment draining Amundsenisen at 2200 m
a.s.l. and merging with Ritscherflya at about 1200 m a.s.l. The bed
topography is known only at a few profiles where radar soundings give
depths of 300 m at shallow points down to 1000 m in deep channels
(Näslund, pers. comm., 2003; Hedfors, 2002). Velocity measurements from
SWEDARP 1987-1995 expeditions combined with remotedly sensed
information (SPOT panchromatic imagery) show that ice movements vary
between 30-70 m a-1 along the main flow direction (Hedfors, 2002). In the
upper part near Amundsenisen the glacier is well confined by the nunataks
of Mathiesenskaget and Sumnerkammen in the northeast and southwest
respectively. Further downstream, at about 1200-1400 m a.s.l., the extent of
the ice flow is limited eastward by Bonnevie-Svendsenbreen, while in the
west an adjacent thin slower moving ice sheet limits the lateral spreading of
Kibergbreen. Accumulation rates upstream Kibergbreen are not well known,
but based on ground penetrating radar (GPR) and snowpit studies, values are
less than 0.05 m w.e.a-1 (Richardson-Näslund, 2001; Richardson et al., 1997;
Näslund et al., 1991). The study on Kibergbreen aided the development of a
thermodynamical model (Paper III) also mentioned in the site description of
Bonnevie-Svendsenbreen above. In addition, a detailed ice surface velocitiy
and ice geometry survey was carried out to serve as input parameters for
later ice flux calculations (not presented in this thesis).
6.4. Plogbreen, Antarctica
Plogbreen (73q02'S, 13q25'W) is a relatively small ice stream located on the
continental margin of Antarctica, draining the Ritscherflya ice field situated
250 km inland. It covers approximately 20-25 km in width and 60 km in
length from the higher ground at Herculessletta to the grounding line at
Riiser-Larsen Ice Shelf and has a surface area of 1420 km2. The ice surface
decreases from 750 m a.s.l. at the ice divide on Herculessletta (near
Ritscherflya) to about 150 m a.s.l. at the grounding line 10-15 km northwest
of Vestfjella mountain range. The most prominent features in this area are
the nunataks of Basen and Plogen, which constrain the ice flow through a
channel-like section at about 200-250 m a.s.l. Previous unpublished radar
sounding measurements between Basen and Plogen indicate an
overdeepening in the southwest reaching ice depths of 1000 m with the base
below sea level (Näslund, pers. comm., 2003). Ice surface velocities have
previously been measured to reach a maximum of ~100 m a-1 in the same
area (Holmlund et al., 1989). Crevasses in the area are mainly associated
34
with the grounding line and restricted areas with relatively steep slope
gradients. However crevasses are also found in large flat areas completely
lacking topographic undulations. The snow accumulation rates varies from
0.25 m w.e.a-1 on Ritscherflya to 0.40 m w.e.a-1 on the Riiser-Larsen Ice
Shelf (Isaksson and Karlén, 1994), while average values for Plogbreen are
estimated to range between 0.32-0.34 m w.e.a-1.
35
7. Results
The following is a brief presentation of the results from the papers included
in the thesis. First are the results from the methodology testing on an
isothermal control glacier (i.e. Storglaciären) to: (i) evaluate the capacity of
the force balance model in a spatial sense (Paper I); (ii) examine the
sensivity of the model to pick up both spatial and temporal changes in ice
dynamics (Paper II). However, outlet glaciers and ice streams in polar
regions are generally polythermal, which motivates the modeling of
thermodynamical conditions of the ice. The third section presents results
from a modeling experiment carried out on Bonnevie-Svendsenbreen and
Kibergbreen to examine the importance of internal heating of the ice in
relation to Glen’s flow law (Paper III). The final two sections (Paper IV and
V) describe the results from applications of Glen’s flow law in combination
with force balance calculations and thermodynamic modeling to estimate inand outfluxes of two polar glacial systems (i.e. Bonnevie-Svendsenbreen and
Plogbreen).
7.1. Testing the model for spatial elements (Paper I)
Original title: Investigating the ratio of basal drag and driving stress in
relation to bedrock topography during a melt season on Storglaciären,
Sweden, using force budget analysis
The ice dynamics of a glacier is governed by the balance between resistive
and driving forces. The driving stress is a function of surface slope, ice
thickness, ice density and gravitational acceleration, while the resistive
forces can be partitioned into stresses along various deformation planes and
frictional drag between the ice and the bed. In particularly, the friction
between the ice and the bed, or basal drag, is known to highly control the
behaviour of the glacier. The resistive forces can be explained in terms of
deviatoric stresses and applied to the general flow law for ice (Glen, 1952).
In Paper I, we investigate the ratio between basal drag and driving stress,
Wb/Wd, on Storglaciären, Sweden, to explore the impact of bedrock
undulations upon glacier flow. We try to explain the basal dynamic situation
by force balance calculations using the isothermal block-flow model fed by
36
input data from field observations (Van der Veen and Whillans, 1989). The
glacier was assumed to be isothermal because the precence of the thin,
slightly cooler, surface layer (shown by Pettersson, in prep) was shown to
not affect the calculated stress distributions. Surface velocity data were
collected using DGPS surveying of a stake net (63 stakes - represented by
black dots in the upper right inset of Figure 7:1) and information on ice
geometry was transferred to a 10 m grid from earlier radio-echo soundings
and digitized surface maps.
The pattern of the calculated ratio of basal drag and driving stress, Wb/Wd,
shows a rythmical position of relatively high and low basal drags on the
stoss and lee sides, respectively, of the bedrock thresholds (Figure 7:1). It
was shown that the ice acceleration across the major threshold at x§22350 m
is induced by raised stresses on the stoss side and governed by low basal
drags on the lee side. Low friction between the ice and underlying bedrock
or till may develop as a result of lubrication (Iverson et al., 1995). Jansson
(1997) pointed out that the hydrological drainage pattern changes from
englacial upstream this threshold to subglacial on the downglacier side. This
can explain the development of a local "slippery spot" downstream at
x§22400 m found in this study, suggesting a "pulling" of ice over the ridge.
Hence, the spatial variations in the surface dynamics observed on
Storglaciären seem to be a reflection of the basal stress conditions shown by
our force balance calculations.
86
85
high
84
low
high low
Figure 7:1 Calculated ratio between basal drag and driving stress, Wb/Wd, for the
lower part of Storglaciären. Values >1 (high) indicate that the basal drag exceeds the
driving stress while values <1 (low) indicate that the basal drag is lower than the
driving stress. Earlier borehole study sites are marked with purple dots accompanied
by drilling year (84, 85 and 86). Coordinate units are in meters.
37
In addition to these calculated dynamic properties, the study gave an
insight in the ability of the method to model ice flow: (i) repeated
calculations for several datasets of surface velocity, covering different time
periods, show similar stress distributions, although with interesting
differences (Hedfors, 2002); (ii) high values of Wb/Wd are found towards the
valley walls where the glacier is frozen to the bed (Pohjola, 1993); (iii) areas
of low values of Wb/Wd align well with areas where the highest rates of basal
sliding, and even extrusion flow, have been measured, i.e. at borehole 84
(Hooke and others, 1987); and (iv) the minima in the calculated Wb/Wd is
found in an area where the water drainage system changes from being
englacial to becoming subglacial (Jansson, 1997). These facts (i-iv) help to
establish confidence in the used data collection technique as well as in the
force budget method.
7.2. Testing the model for temporal elements (Paper II)
Original title: Seasonal variations in the ice dynamics of Storglaciären,
Sweden
One of the interesting problems on Storglaciären concerns the temporal
variations in the dynamics of the glacier, i.e. strain and stress, across the
major bedrock threshold in the lower part of the glacier. Previous studies
(e.g. Hooke et al., 1989) indicate a seasonal variation in both vertical and
horizontal surface strain rates in relation to subglacial water pressures
measured at the bedrock threshold. Paper II extends the previous work by
presenting a hypothesis to explain the observed seasonality in the surface
signals by investigating the basal stress conditions across the threshold. We
use the same ice geometry data and methodology as in Paper I, this time to
estimate the seasonal ratio between basal drag and driving stress. Surface
velocity data from 3 different seasons were collected during a DGPS survey
of 42 stakes in 2000-2001.
The calculation provides detailed information on the spatial and temporal
changes in surface strain rates (Figure 7:2) confirming the previous studies
(Hooke, 1989). It also reveals interesting anomalies in the basal stress field
obviously controlling the temporal changes observed at the surface (Figure
7:3). The bedrock threshold in combination with meltwater availability seem
to be the key factor in controlling the basal drag in such that it in situ causes
low basal drags and compressional flow in spring and high basal drags with
extensional flow in winter and summer. The "sticky" conditions in winter
38
6
x 10
13
0
7.5348
12
75
Y (m)
Y (m)
1175
2.22
2.23
X (m)
0.005
12
50
7.5344
25
12
7.5343
12
7.5342
75
12
50
7.5341
1275
2.24
2.25
2.21
7.5341
b
1350
2.22
2.23
X (m)
4
x 10
-0.01
13
00
25
13
7.534
2.2
0
-0.005
00
12
2.21
25
12
7.5342
a
1350
7.534
2.2
00
12
00
13
5
132
0.01
1175
50
12
0
130
7.5341
25
7.5342
5
7.5344
7.5343
12
12
7
0.015
1250
1225
12
00
7.5345
1200
1200
5
122
7.5343
0.02
7.5346
1175
00
12
0.025
1325
130
0
1275
7.5347
12
25
12
00
7.5345
7.5344
x 10
7.5348
12
50
7.5346
7.5345
Y (m)
127
5
7.5347
12
00
7.5349
1325
1300
7.5348
1250
7.5346
6
x 10
0
122
5
7.5347
7.5349
1325
12
00
6
7.5349
2.24
132
5
2.25
2.21
2.22 2.23
X (m)
4
x 10
-0.015
c
0
135
7.534
2.2
2.24
2.25
-0.02
4
x 10
Figure 7:2 The difference in longitudinal surface strain rates, H xx, between annual
mean and the seasons: a) winter; b) spring; and c) summer. Positive values indicate
an increase in H xx (longitudinal stretching) and negative values indicate a decrease
in H xx (longitudinal compression). In winter, the deformation does not change
significantly from the annual mean values, while in spring, a clear signal of
compressional flow is noted over the entire threshold. In summer, the flow becomes
more or less extentional in that same area.
6
6
x 10
7.5349
1325
1300
1275
7.5348
1250
12
2
120
2.21
Y (m)
Y (m)
1175
2.22
2.23
X (m)
7.5341
a
2.24
2.25
4
x 10
7.534
2.2
1350
12
75
2.21
2.22
2.23
X (m)
2.24
13
7.5341
b
2.25
4
x 10
7.534
2.2
0.8
50
12
0
135
25
12
7.5342
12
75
13
00
132
5
0.7
00
132
0
135
2.21
1
0.9
1200
132
5
00
12
0
00
13
7.5344
7.5343
120
75
12
1.2
12
00
1175
1175
7.5342
0
1.3
12
50
1.1
00
12
25
12
1.4
75
7.5345
50
12
5
12
12
25
7.5346
7.5344
7.5343
12
25
1.5
1325
1300
7.5347
00
12
7.534
2.2
5
25
7.5341
00
12
12
7.5342
50
7.5346
0
x 10
7.5348
75
12
7.5345
122
1300
12
5
1200
7.5343
12
7.5347
7.5345
7.5344
7.5349
1325
7.5348
7.5347
7.5346
6
x 10
Y (m)
7.5349
2.22 2.23
X (m)
c
0.6
5
2.24
2.25
0.5
4
x 10
Figure 7:3 The ratio between basal drag and driving stress, Wb/Wd, calculated for the
three seasons: a) winter; b) spring; and c) summer. Generally, balanced conditions
are found at the bed for all time periods as indicated by values near zero. However, a
zone of strong negative ratio is found in spring, extending laterally at the stoss side
of the threshold and indicating low basal drags. In summer, this zone of low friction
disappears, instead a negative value is found on the lee side.
39
can be explained by the lack of lubrication from meltwater and closing of
subglacial channels. In spring, meltwater enters a poorly developed drainage
system generating high water pressures at the threshold accompanied by
glacial uplift and a "slippery" plug flow. Another source for temporal
lubrication is discussed, as regelation processes seem to fit the model
outcome. In summer, a steady input of meltwater is sufficiently drained
through a successively more developed subglacial drainage network, thus
maintaining a relatively low subglacial water pressure and high basal drags
over the threshold.
All together, these variations in spatial and temporal extent, illustrates the
harmonics in which the glacier shifts its behaviour from extensional flow in
winter and summer to compressional in spring. Opposite to what initially
was believed, the threshold stoss-side temporarily exhibits low basal drags
suggesting that flow related obstacles do not neccessary cause sticky spots.
7.3. Testing a thermodynamical model (Paper III)
Studying the effects of strain heating on glacial flow within outlet glaciers
from the Heimefrontfjella Range, D.M.L., Antarctica
Glacier ice warm up under the influence of geothermal heat fluxes and
frictional heating, or strain heating, within the ice due to ice crystal
deformation processes. Higher temperatures cause glacier ice to flow faster
due to a reduced stiffness, or lower viscosity. The strain heating is an
important component in glaciers of great thicknesses and steep slopes and,
thus, needs to be considered in flow models of ice in cold environments, e.g.
ice streams in polar regions. With Paper I and II in mind, where an
isothermal model were used, Paper III presents a way to establish
temperature profiles for use in modeling of ice flow in larger outlet glaciers.
Our study uses a one-dimensional numerical thermodynamic algorithm
based on the general equation for heat transfer given a set of input
parameters such as ice depth (h), surface slope (Į), surface temperature (T)
and estimations of accumulation rate ( a ). An experimental study of
Bonnevie-Svendsenbreen and Kibergbreen, DML, Antarctica, showed that
Glen’s flow law for ice could not explain observed values of surface
velocities when strain heating was excluded. The values were too low. By
incorporating the effects of strain heating along the flowline of the two
glaciers, the calculation returned higher values of surface velocities which
also matched the measured values (Figure 7:4).
Also, it was found that relatively short scale temporal and spatial steps in
basal topography are sufficient to drive the ice flow into a positive feedback
40
loop as long as the bedrock step produces a stress that overcomes the
advection of cool ice from the surface. In this case, where surface
temperatures are –25 qC stresses of 0.4 MPa are sufficient to drive the base
of the ice to the melting point within 100 years. These results highlight the
importance of heating from various sources within the ice and, to some
extent, explains the mechanism behind fast flowing ice streams. The
subsequent aim is to use this strain heating algorithm to establish better
temperature profiles for use in following ice stream studies.
-25.00
-20.00
-15.00
-10.00
-5.00
0.00
0
0.0
(S-S)
Ice depth (m)
a
40.0
60.0
80.0
0
(80)
(60)
20
BSB
(40)
b
100
100
100
200
200
200
300
300
300
400
(40)
(S-S)
(60)
BSB
(80)
400
400
-25.00
-20.00
-25.00
-20.00
-15.00
-10.00
-5.00
0.00
-15.00
-10.00
-5.00
0.00
Ice temperature (°C)
0
0.0
0.0
20.0
20.0
0
c
200
Ice depth (m)
20.0
0
40.0
Ice velocity (m / a)
40.0
60.0
60.0
80.0
80.0
100.0
0
(30)
(S-S) 10 (20)
(35)
d
200
200
400
400
400
600
600
600
800
800
800
(S-S)
1000
-25.00
-20.00
10 (20)
-15.00
(35)
(30)
-10.00
Ice temperature (ºC)
-5.00
1000
0.00
1000
0.0
20.0
40.0
60.0
Ice velocity (m / a)
80.0
100.0
Figure 7:4 The thermal (a and c) and ice speed evolution (b and d) from the model
using input data from Bonnevie-Svendsenbreen (a and b) and Kibergbreen (c and d)
respectively. The distributions shown are products of the steady-state thermal profile
(S-S) using no strain heating, and with strain heating of the steady state profile after
20, 40, 60, and 80 years for Bonnevie-Svendsenbreen and 10, 20, 30, and 35 years
for Kibergbreen. The circles in b and d indicate the observed average surface speeds
on the respective glacier.
41
7.4. Mass flux of Bonnevie-Svendsenbreen (Paper IV)
Original title: Investigating the potential to determine the upstream
accumulation rate, using mass flux calculations along a cross-section on a
small tributary glacier in Heimefrontfjella, D.M.L., Antarctica
The vast areas of Antarctica provide a delicate problem of determining the
incoming mass of an individual glacial system. Large uncertainties reside in
the spatial variability observed in accumulation rates (Oerter et al., 2000;
Isaksson et al., 1999; Richardson and Holmlund, 1999) as well as in
determining the location of the ice divides outlining the drainage areas. This
paper has a twist to the usual scientific question of estimating in- and out
fluxes of a given glacial system: how well can we estimate the incoming ice
flux by calculating the ice outflux through a well defined cross-section? We
test this by comparing calculated ice flux out from Bonnevie-Svendsenbreen
with the measured accumulation rate integrated over the well defined
catchment area in the Sivorgfjella plateau (Figure 7:5).
Sivorgfjella
Plateau
N
5 km
Figure 7:5 Bonnevie-Svendsenbreen with the Sivorgfjella Plateu and the
investigated cross-section A-A' (thick dashed line).
The ice flux is calculated using the ice dynamical properties from the ice
temperature model developed in Paper III and the distribution of forces
calculated using the force budget model developed and used in Paper I and II
(Figure 7:6 and 7:7). Input data includes velocity data of the glacier surface,
combined with ice thickness measurements.
The result is an accumulation rate on the Sivorgfjella plateau of 0.50±0.05
m.w.e.a-1 which is shown to correspond well with the accumulation rate of
0.50±0.30 m w.e.a-1 recorded by GPR work in the area (Richardson and
others, 1997). We therefore find the balance flow method, in combination
with the force budget technique, and ice temperature modeling, to be a
useful tool for studies of mass fluxes in a catchment area. The calculated
42
high accumulation rate shows the effect of orographic enhancement on
accumulation in montane areas in Antarctica.
Y-axis
8000
A
c
b
a
8000
7500
7500
7000
7000
6500
6500
6000
6000
5500
5500
5000
5000
Y-axis
11000 11500 12000 12500
4500
4500
11000 11500 12000 12500
X-axis
A'
11000 11500 12000 12500
Figure 7:6 a) The calculated distribution of depth integrated effective strain (a-1), b)
basal temperature (°C), and c) basal drag (kPa) as result from the ice temperature
and force budget modeling.
Y-axis (m)
Altitude
(m a.s.l.)
A'
Altitude
(m a.s.l.)
5000
5500
6000
6500
7000
1600
1400
1200
1000
1400
1200
a
1200
1000
A
4500
1400
1200
5000
5500
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6500
7000
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b
4500
1400
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5500
6000
6500
7000
1600
1400
7500
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1000
Altitude
(m a.s.l.)
4500
1000
7500
1600
c
4500
1400
1200
5000
5500
6000
6500
7000
1000
7500
Y-axis (m)
Figure 7:7 The distribution of a) ice velocity (m a-1), b) effective stress (Pa), and c)
ice temperature (°C), using isolines of 2.5° equidistance, in the cross-section A-A’.
43
7.5. Mass flux of Plogbreen (Paper V)
Original title: Ice flux of Plogbreen, a small ice stream in Dronning Maud
Land, Antarctica
In 1988, the Swedish Antarctic Research Programme (SWEDARP) initiated
a long-term mass balance program in the Vestfjella-Heimefrontfjella (VH)
area, DML, Antarctica. The general aim is to determine whether this part of
the Antarctic ice sheet is in balance with present climate. Paper V
contributes to this aim by studying the transfer, or flux, of ice in- and out
from Plogbreen, one of the glacial systems near Vestfjella (Figure 7:8).
73qS
A
Basen
N
GATE
A'
2
Plogen
20'
10 km
14q W
ice divide
1
13q
30'
Figure 7:8 Overview of the study area. The rectangle at the cross section shows the
approximate area/gate of the DGPS and GPR surveys. The dashed line is an outline
of the catchment area. The dotted line represents the trajectory path used in the
modeling of temperature with ice depth, where 1 and 2 stand for different stages in
the model algorithm. Included in the northwest corner is also the approximate
location of the grounding line. Swarms of small triangles indicate zones of observed
crevasses.
We compare ice outflux, ‡out, through a cross-sectional gate with ice influx,
‡in, from the upstream cathment area. In order to find the outgoing mass,
collected field data from DGPS and GPR surveys (Figure 7:9a and b) are fed
into an ice flow model that integrates the ice surface velocity over the ice
44
geometry. The model is based on Glen's flow law for ice in combination
with thermodynamic modeling (Figure 7:9c) and ice dynamical calculations,
or force balances at the gate (Hedfors, 2002; Van der Veen and Whillans,
1989). Basal drag, depth-integrated strain rates and modeled ice flow are
presented in Figure 7:10.
These calculations provide a total outgoing volume of 0.55r0.05 km3 a-1.
The drainage area of Plogbreen was outlined using Landsat TM imagery in
combination with AVHRR-based photoclinometry derived digital elevation
models, and estimated to 1420r300 km2. This gives a total inflow of
0.48r0.1 km3 a-1 from the upstream area, given the accumulation rates from
a
16000
b
18000
302520
40
45
50
55
60
65
70
75
80
85
35
40
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15
15
Ice depth (m)
25
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x (m)
3000
0
200
0
1000
2000
x (m)
3000
1100
-20
-18
-16
-14
-12
Ice temperature (ºC)
-10
-8
Figure 7:9 Collected field data and modeled vertical temperature profile: a) is DGPS
derived horizontal surface velocities (m a-1) in x-direction (main flow direction).
Arrows show relative velocity vectors projected from the actual 40 stake positions
(black dots); b) shows GPR recorded ice depth (m). Note that the figure uses a local
coordinate system rotated approximately 160q in comparison with the frame outlined
in Figure 7:8. c) is a calculated ice temperature profile using one stage of steady
state modeling (location 1 in Figure 7:8) followed by a stage of temperature
evolution due to higher driving stresses (from location 1 to 2 in Figure 7:8).
45
a
A
A'
b
c
Figure 7:10 a) Horizontal projection of basal drag, Wbx (kPa), with the cross section
marked (dashed line) at x = 1500 m (A - A'). Ice flow is from top to bottom. b)
Depth integrated effective strainrates, H e (a-1), and c) horizontal velocity with depth,
Ux (m a-1). The effective strain rate (b) and velocity field (c) is shown looking
upstream with a depth exaggeration (u4) for better visualization.
previous ice core records (Isaksson and Karlén, 1994). By only considering
the median values of the flux components, the Plogbreen system can be said
to be in a negative state where the mass leaving the system is greater than
that of incoming mass (0.55>0.48). This is also suggested from ice core
records showing a falling trend in accumulation (-25 %) over the past 70
years (Isaksson, 1994). More interesting is the glacier's proximity (~10 km)
to the grounding line and the sub sea-level overdeepened trough (~800 m
below sea level) observed towards the Plogen massif. This environmental
setting may allow for detection of signals of global warming. Thus, as a
speculative comment, we argue that the indications of a negative mass
balance can be explained by the recent rise in the global sea-level, as it is
likely to induce glacier acceleration due to a reduction in resistive forces at
the site of the gate. The argument is supported by observations of recent ice
shelf break-up at the Riiser-Larsen Ice Shelf in front of Plogbreen. However,
46
more studies of similar kind in this area are required to fully understand the
processes that affect the glacier response to climate change. Additional data
and/or extended field campaigns remain to confirm or discard the overall
state of the mass balance in the Vestfjella-Heimefrontfjella area.
47
8. Discussion
8.1. Force budget tests on Storglaciären
In the perspective of previous work carried out on Storglaciären, my studies
in Paper I and II add detailed information on the spatial and temporal
variations in ice dynamics in relation to bedrock geometry as well as
seasonal influences upon glacier flow. It showed that the basal stress
conditions governing the dynamic situation on the surface can be estimated
via force balance modeling. The strain and stress fields provided by the
calculations fits the general picture of ice flow over an undulating bedrock
topography when tuned to produce an outflux similar to a zero net balance
year, i.e. 1.4·106 m3 a-1, at the site of the threshold (Grudd and Bodin, 1991).
In addition, it was capable to discern details in the basal stress conditions not
previously known. These details could to some extent be coupled to both
spatial features and temporal events that had been observed previously. The
results from the previous borehole studies turned out to match what we saw
in the force balance calculation, at least in a spatial and relative sense. And
this is where the strength of the force balance method crystalizes; it may not
produce the exact magnitudes of stresses with depth due to imperfect
modeling, however, both the spatial and temporal patterns of the force
balance terms pinpoint the relative differences in the ice dynamics. A
drawback is found in the fact that H xz and H yz are neglected in the force
budget calculation of effectice surface strain rates. These components are
most important at the margins as shown by Pohjola (1993), however in the
center at the threshold stoss side, they were calculated to a fraction of for
example H xx ( H xz <0.001 a-1 at borehole 85). A good test of their importance
could not be carried out in this thesis, however, a simple test shows that the
pattern of basal drag is slightly amplified when including average strain rates
of up to 0.005 a-1 in the xz-plane near the threshold.
The outcome from this work has not only explained some of the peculiar
features of Storglaciären, but also raised many new questions: How does
longitudinal coupling relate to temporal variations in turning of flow? How
important are the bedrock thresholds in relation to regelation processes? Is
the temporal basal friction loss on the threshold stoss-side induced by
48
meltwater from underdeveloped subglacial channels or by meltwater from
processes of regelation? An extended and improved study of force
distributions from a larger area with complementing borehole information
could give answers to questions like these. Other studies in progress (e.g.
Jansson, pers. comm., 2003) studies the water drainage from the
accumulation area through the overdeepenings. Such work would be useful
for further dynamical studies if combined with a three dimensional view of
the force distribution. Recent information from radar surveys (e.g. Petterson,
pers. comm., 2003) has mapped the cold surface layer and also revealed
properties on the englacial drainage pattern. This is important knowledge
that also would guide further force balance studies by supplying a basis for
better viscosity calculations and enable us to understand the role of lateral
drag in relation to H xz and H yz as well as the mechanism behind changes in
basal sliding. The work from Storglaciären reported in this thesis is in that
sense a step towards understanding the details in the ice dynamics, but far
from giving the complete picture. It does however meet the demands for
establishing basic tools for similar investigations of ice flow in polar regions
where the background information is much more limited.
8.2. Force budget applications in Antarctica
The force balance technique was used for two glacial systems in Antarctica,
Bonnevie-Svendsenbreen and Plogbreen, to estimate the transfer of ice mass
within respective system. Thus, another step is taken from mapping the ice
dynamics to actually determining the ice flux through a gate. With the
experience from Storglaciären, much of the strain and stress fields obtained
from these Antarctic systems were recognized as logical models for the true
physics, however the limited data made available for these glaciers put
constraints on the results. We encountered difficulties in estimating mass
influx due to spatial variability in ice accumulation and fuzzy catchment
boundaries. The problem was partly overcome by using remotedly sensed
imagery from a combination of satellite based platforms, e.g. AVHRR-based
photoclinometry derived digital elevation models, and surface features from
RADARSAT-1 synthetic aperture radar (SAR), Landsat TM and feature
tracking from sequential SPOT 10 m panchromatic data. The
thermodynamical model accompanied with its assumptions is incoorporated
to account for the polythermal character of the glaciers. The information on
ice depths was a problem in the case of Bonnevie-Svendsenbreen due to
sparse measurements (spatial cover) and consequently supraglacial bedrock
topography had to be used to interpolate data into a map of the bedrock
topography. On Plogbreen, the problem became even clearer as weak GPR
signals from the bed were hard to interpret. A few previous studies did
49
however confirm both our velocity- and ice depth measurenments on
Plogbreen.
In context with the initial summary of messages from Antarctica, I argue
that the studies in Paper III, IV and V, has contributed to the understanding
of the ice dynamics of this vast continent. Because Paper III suggests that ice
driven by high stresses may experience a substantial temperature increase
through processes of strain heating, founding a feedback loop of
successively increasing velocities with a subsequent rise in ice discharge to
the oceans. If the temperature causes basal melting in glaciers far inland as a
consequence of higher surface temperatures and/or lower cooling effects
from annual snow accumulation we may see an accelerated discharge of
several ice streams across the entire continent. As exemplified by studies
(e.g. Frezzotti et al., 2000 and Huybrechts and De Wolde, 1999) basal
melting is a mechanism to ice sheet collapse. This is prominent (as seen
today) in glacier systems near the grounding line, however, were the basal
melting conditions to propagate further upstream, we would soon notice
accelerated flow in many systems.
Raised air temperatures and eustatic changes of the sea level also relate to
the mechanism of ice shelf break up (Wingham et al., 1998). Field
observations from our SWEDARP expedition 2002/03 gave an indication of
initial (local) ice shelf break up downstream Plogbreen. This is interpreted as
a signal of a change taking place, perhaps somwhere within the Plogbreen
ice stream. As it turned out, the study on Plogbreen presented in Paper V
also suggested an imbalance between ice mass input and output, an
imbalance that indicated a higher discharge compared to accumulation.
Plogbreen has its base well below sea level and is as such sensitive to
eustatic changes. We argue that the indications of a negative mass balance
can be explained by the recent rise in the global sea-level, as it is likely to
induce glacier acceleration due to a reduction in resistive forces at the site of
the gate.
Paper IV found a high potential in estimating the transfer of ice within
glacier systems by using outflux through a gate as a gauge for upstream
accumulation. The errors in the accumulation rate associated with the force
balance calculations at the gate were less than those associated with an
integration of accumulation rates based on GPR surveys in the catment area.
This is due to high spatial variability associated with GPR derived
accumulation rates, an uncertainty that is not applicable in the force balance
calculations.
Considering the study on Plogbreen, this suggests that also here, the error
in the mass input component is most likely to be the largest. The difference
in uncertainty was estimated to twice as large for the influx in the catcment
area (±23.1 %) compared to the transfer through the gate (±10.5 %). The
uncertainty associated with the modeled temperature is excluded in the
50
outflux error because repeated simulations of various configurations did not
produce much difference in the temperature profiles. In turn, small variations
in the temperature profile had non-significant effects on the final flux output.
The total heating generated within Plogbreen caused a temperature increase
from -18 qC at the surface at the ice divide to about -9 qC at the deepest parts
of the glacier at the gate. It is reasonable to believe that this is near a true
value, since the grounding line (10-15 km downstream the gate) generates
massive surface crevassing demonstrating an equally massive acceleration in
ice flow. Hence, it is hypothized that the major part of the ice bed starts to
slide on a lubricating waterfilm in this area. If the basal conditions had
reached pressure melting point upstream the gate, we would have expected
to see a similar phenomenon here.
In addition, no meltwater has been observed at the glacier surface
upstream the gate, and the calculated basal drag is considered to be too high
(200-300 kPa), in comparison with values of calculated driving stress
(equally high), to represent processes of major basal sliding. This does not
exclude the possibility of parts of the bed being subject to basal sliding at the
gate due to the intrusion of ocean water together with hot spots generated by
pressure anomalies against bedrock outcrops. This can not be accounted for
in the thermodynamic model. Thus, the error range for this calculation does
not outrule the possibility of this area being in a state of balance, which is
the case with the nearby ice stream Veststraumen (Holmlund et al., 2003). In
that perspective, the reliability of the results from Plogbreen can be
questioned. To attack the problem, I would therefore focus on the influx
component in future studies by making extensive use of satellite imagery to
improve the delimitation of the cathment area together with interferometry
based mapping of flow components in, and adjacent to, the major bulk flow.
The geographical location of Plogbreen adds an interesting aspect on the
interpretation of the mass balance result. Would it be influenced by
conditions prevailing in the West or would it respond to the changes
dominating in the East Antarctica? A balanced system, or even a slight
positive mass balance would place Plogbreen and its dynamic character
together with glacier systems in the east. A negative system could mean that
it is subjected to changes similar to those taking place in the West Antarctic
Ice Sheet, the Amundsen Sea sector and the Antarctic Peninsula. Since
Plogbreen is situated on the borderline between these geographically and
dynamically different parts, the answer is not straightforward.
Some regional differences in the general circulation patterns are evident
over and around Antarctica. In West Antarctica, high moisture fluxes (>150
mm w.e.a-1) on a wide front reach far inland over the relatively low altitudes
beyond 80qS accompanied with high accumulation rates (Budd et al., 1995).
The extremes (>450 mm w.e.a-1) are found on the Antarctic Peninsula which
is also strongly affected by frequent cyclons from northwest. King and
51
Harangozo (1998) report that during the past 20 years, the Antarctic
Peninsula has experienced pronounced warming. There are two ocean gyres,
the Weddell Gyre and the Ross Gyre, in this region that may influence the
rate of moisture exchange between the ocean and atmosphere. In the East
Antarctic region, moisture fluxes of up to 150 mm w.e.a-1 are concentrated to
coastal basins in Wilkes Land, around Amery Ice Shelf and Enderby Land,
below 70qS (Budd et al., 1995). The high altitudes of the ice sheet in this
region seem to prevent the transfer of moisture further inland. In addition,
Comiso (2000) reports a temperature decrease at a number of weather
stations on the coast and plateau of East Antarctica. The offcoast counterclock ocean circulation is not affected by any gyres similar to those in the
west. Thus, for any changes in the atmosphere or ocean circulation, we
would expect to find the most dramatic ice dynamical changes in the west
due to higher rates of exchange between atmosphere and cryosphere. This is
also exemplified by the research results from previous studies (e.g. Shepherd
et al., 2001; De Angelis and Skvarca, 2003) according to whom negative
mass balances dominate West Antarctica. The state of the mass balance in
East Antarctica is not yet settled although some studies indicate general
positive fluxes (e.g. Rignot et al., 2002; Rignot and Thomas, 2002).
The moisture flux upstream from the Vestfjella-Heimefrontfjella area,
DML, rarily exceeds 50 mm w.e.a-1 even though it is subjected to frequent
cyclon activity originating over the Weddell Sea. Locally, values of 500±50
mm w.e.a-1 are associated with orographic enhancement on accumulation in
montane areas as shown in Paper IV. Interestingly, studies report a trend of
decreasing accumulation in the coastal areas (e.g. Isaksson and Karlén,
1994) due to cooler winter and fall temperatures or displacements of
cyclogenic trajectories. The general temperature trend is positive for the last
~100 years increasing by 1.8 qC in the coastal areas and 0.8 qC on higher
altitudes (>2500 m) further inland. These values fall between the average
temperature increase over the Antarctic Peninsula of 2.5 qC over the past 50
years (Vaughan and Doake, 1996) and the general Southern Hemisphere
warming of 0.5 qC since the end of the 1800s (Isaksson and Karlén, 1994).
This means that precipitation is decreasing and temperatures are rising at the
site of Plogbreen. A natural glacier response to this would generate a
negative mass balance over time, similar to the conditions measured in
numerous places in West Antarctica. It is therefore tempting to suggest that
the negative mass balance of Plogbreen shown in my studies represents an
early signal of changes to come - changes of the same nature as those now
occurring in many places in West Antarctica.
To round up this discussion, it has been shown that the model is capable
of estimating ice fluxes in polar regions, its strength demonstrated by the
relative low errors produced in the outflux component at the gate.
Information on ice dynamical conditions have been successfully revealed
52
and improved by thermodynamical modeling. Preferably, the force balance
model should be used as a tool for outflux estimations with a strong support
from both field observations and satellite imagery analysis of environmental
properties. For future studies, I recommend: i) further applications of this
methodology on polar glaciers (e.g. Veststraumen, DML, Lomonosovfonna,
Svalbard, and outlet glaciers from the Greenland ice sheet); ii) comparisons
of results from this model with results from other techniques applied on the
same glacial system (e.g. interferometry based investigations); iii) further
improvements of the model used here (e.g. replacing the force budget blockflow scheme with a layered scheme in terms of numerical calculation
approach); and iv) additional evaluation of the thermodynamic model in
relation to basal ice conditions and flow velocities (e.g. by testing modeled
temperature evolution for glaciers of known basal conditions).
53
9. General conclusions
This study aimed to collect, process and analyze geophysical and remotelysensed data of glacier movement and geometry, and with an improved
modeling technique, use this data to establish spatial and temporal patterns
of ice dynamics to estimate ice mass fluxes of glaciers in polar regions.
Geophysical data (i.e. surface velocities and ice geometry) was collected,
processed and analyzed for 4 glacial systems: Storglaciären, BonnevieSvendsenbreen, Kibergbreen and Plogbreen, by using combinations of in situ
DGPS and GPR techniques together with remotely-sensed data from mainly
SPOT panchromatic and RADARSAT SAR imagery. A methodology of
combining force balance calculations with calculations of thermodynamical
properties of ice was incorporated in a model to calculate ice flow via
inversion of the flow law for ice. Given the gathered geophysical and
remotedly sensed data, the developed model returned valuable information
on both spatial and temporal patterns in the ice dynamics of the different
case studies.
In particularly, the calculated basal stress fields from Storglaciären
provided an explanation for signals observed in surface deformation, both
spatially and temporally. Here, the spatial distribution of forces is strongly
correlated to bed topography and the temporal variations in the calculated
stresses are believed to relate to changes in basal water pressures induced by
seasonality. A few conclusions can be listed:
‰
‰
‰
‰
The strength of the force balance method resides in its capability of
discerning relative differences in both spatial and temporal patterns
of glacier flow.
The force balance calculation from Storglaciären shows a repetitive
pattern in the ratio between basal drag and driving stress, as the ice
flows over bedrock ridges and into overdeepenings.
Basal conditions are on average found to be stickier on the stoss side
of the ridges and more lubricated on the top and lee sides of the
ridges.
Borehole studies indicating high basal sliding rates, and even local
extrusion flow, were found to be located in the same area as the
lowest calculated basal drags. Relatively low basal sliding rates
54
‰
‰
(measured in the boreholes) corresponded to areas where we found
high basal drags.
The area of the lowest calculated basal drag (for the summer period)
is situated where the water drainage system changes from englacial
to subglacial.
However, the bedrock threshold in combination with meltwater
availability seem to be the key factor in controlling the basal drag in
such that it on the stoss side causes low basal drags and
compressional flow in spring and high basal drags with extensional
flow in winter and late summer.
The thermodynamical experiments on Bonnevie-Svendsenbreen and
Kibergbreen clearly showed that Glen's law could not acount for the total ice
surface velocities without a component of internal strain heating. The major
discoveries are summarized as follows:
‰
‰
‰
By incorporating the effects of internal strain heating along a
flowline of both Bonnevie-Svendsenbreen and Kibergbreen, Glen's
flow law returned values of surface velocities which matched the
measured values.
A stress threshold value of about 0.4 MPa - produced by, for
example, a steeper surface gradient - could be enough to drive the
ice flow into a positive feedback loop as long as it overcomes the
advection of cool ice from the surface.
The results show that the base of the ice on BonnevieSvendsenbreen and Kibergbreen reaches melting point if the stress
threshold is exceeded for approximately 100 years.
This highlights the importance of internal strain heating within the ice and,
to some extent, explains the mechanism behind fast flowing ice streams.
The results from the ice flux calculations of the case studies in Antarctica
were successful so far that they provide an estimate of the transfer of ice
within the glacial systems. The attempt to estimate the accumulation rate on
the Sivorgfjella plateau in Heimefrontfjella by using the outflux as a gauge
for influx showed the following:
‰
‰
The total ice flux through Bonnevie-Svendsenbreen is 0.041±0.004
km3 a-1, assuming steady state conditions.
The accumulation rate on the Sivorgfjella plateau is 0.50±0.05 m
w.e.a-1, which is shown to correspond well with the accumulation
rate recorded by GPR work in the area.
55
‰
‰
We find the balance flow method, in combination with the force
budget technique, and ice temperature modeling, to be a useful tool
for studies of mass fluxes in a catchment area.
The calculated high accumulation rate shows the effect of orographic
enhancement on accumulation in montane areas in Antarctica.
The work on Plogbreen in the Vestfjella area, provide the first detailed
investigation of mass flux for this glacial system. The major findings
conclude:
‰
‰
‰
Plogbreen recieves an influx of 0.48r0.1 km3 a-1 and expedites a
total outgoing discharge volume of 0.55r0.05 km3 a-1.
The calculations indicate a negative imbalance where the mass
leaving Plogbreen is greater than that of incoming mass (0.55>0.48).
Considering the ice stream's proximity to the grounding line, the
indications of a negative mass balance can be explained by the
recent rise in the global sea-level since eustatic changes are likely to
induce glacier acceleration due to a reduction in resistive forces at
the site of the gate.
56
10. Acknowledgements
Glaciology has taught me to depend on people. Whether you are dangling at
the end of a rope 20 m down a crevasse, or need assistance in the art of
resizing scientific figures; or you are cooking breakfast on an upside-down
snowscooter at -40 qC; or need help to fill in an application form. Whether
you are stuck in polar bear country with a completely frozen-up gun, or wish
to tag along to a conference; or you are digging yourself out of a tent after
days of snowstorms or formulating a response to a referee. You always need
a hand as a PhD-candidate. By now, of course, that time has passed and I can
manage on my own, or?
The list of people I wish to acknowledge starts with my supervisor, Veijo
Pohjola, who arranged for me to visit places where few men (or women)
ever put his (her) foot. I am sincerely thankful for your great spirits on field
trips to Tierra del Fuego and Spitsbergen, and for opening the door to
Antarctica, where you even let me run the show. You have shown a sincere
interest in research methodology and kept me going through the years,
sometimes practicing shocking approaches. However, "nothing shocks me,
I'm a scientist" (Jones, I., in Lucas, 1984). Likewise, my second supervisor,
Else Kolstrup, has shown great interest and provided help in times of need.
Peter Jansson introduced me to Tarfala Research Station (with a great
supporting staff) and the research traditions on Storglaciären providing
invaluable help in the field as well as at the desk. A generous load of
assistance from Rickard Pettersson is also highly appreciated as well as
guidance and gear supply organized by Per Holmlund. Scientific instruments
were at times provided by Eric Roland at Norweigan Water Resources and
Energy Directorate and John Moore at the Arctic Centre, Rovaniemi, who
also showed a remarkably ambitious level in teaching on field courses on
Storglaciären, usually at times of raging storm winds. Superb GPS-gear
support was given by Lars Gustafson and Annika Ljungberg from Caliterra,
Sweden.
Numerous people have helped me in the field carrying out scientific
measurements as well as daily routine tasks. Credits should be given to Ola
Brandt, Josef Källgården, Tomas Jacobsson and Björn Sjögren for assistance
on Svalbard glaciers, and Håkan Samuelsson for sacrifying a shoulder in the
name of science. Elisabeth Isaksson provided help in the field and solved
logistic issues at the Norwegian Polar Institute. Field work activities in
57
southern Patagonia (indirectly contributing to this thesis) were supported by
great efforts from Keith Bennett, Charlie Porter, Camilla Hansen, Alfredo
Nivaldo Lucero and Gino Casassa. Additional gratitude goes to Josef
Källgården for showing unsurpassed loyalty in the field campains in
Antarctica. Here, I also owe special thanks to the entire crew of SWEDARP
2002/03, honouring the Kiberg Festival Comittee; Mats Nilsson, Mats
Johnsson, and Markus Karasti, for providing safety measures and loud and
clear means of communication.
Personal communications with Jens-Ove Näslund, Ted Scambos, Kjetil
Melvold, Douglas Mair, Jonathan Bamber, Richard Hindmarsh, Tony Payne
and Luca Vittuari have helped to improve the papers published in this thesis.
I wish to thank Regine Hock for passing criticism on my work at "halftime" and especially my final opponent, Niels Reeh together with the
examination comittee, Jon-Ove Hagen, Regine Hock (again) and Stig
Jonsson, for showing interest in my undertaking.
Guidance from all collegues at the Department of Environmental and
Landscape Dynamics is appreciated as well as help from library staff at
Geobiblioteket. Thanks, also to all PhD-candidates at Geocentrum for
inspiring talks and great TGIF sessions.
Grants were gratefully recieved from SSAG (Swedish Society of
Anthropology and Geography), YMER-80, C.F. Liljewalchs travel fund, and
through logistic support from the Swedish Polar Secretariat. Without this
financial support, none of my work would have been possible.
At last, but not least, I am in debt to my family and friends, who endured
long periods of my absence and abnormal behaviour associated with my
curiosity for glaciers. I am particularly impressed by Anna who put up with
my cold feet.
58
11. Summary in Swedish
Kraftbudgetanalys av glacialt flöde
Isdynamiska studier på Storglaciären, Sverige, och
isflödesundersökningar av utlöparglaciärer i Drottning
Maud Land, Antarktis
Syftet med denna studie är att bidra till förståelsen av klimatbetingade
reaktioner i glaciala system genom studier av Storglaciären, Sverige, och
utlöparglaciärerna Bonnevie-Svendsenbreen, Kibergbreen och Plogbreen i
Drottning Maud Land, Antarktis.
Glaciala flöden drivs av spänningar som uppkommer på grund av
gravitationskraften och motverkas av spänningar som utbildas i och med
friktionen mellan glaciären och underliggande berggrund. Detta
spänningsförhållande utgör de dynamiska egenskaperna för allt glacialt flöde
och bestämmer hastigheten i massutbytet genom ett glacialt system. Mitt
arbete använder en kraftbudgetmodell för att beräkna utflödet av is i glaciala
system genom den tredimensionella distributionen av spänningar vid en
geometriskt väldefinerad sektion, eller grind, kombinerat med etablerade
fysiska lagar för isdeformation och termodynamik. Arbetets vikt ligger i
relationen till utbytet som sker mellan vatten lagrat som is i kryosfären
(glaciärer, permafrost etc) och vatten lagrat i hydrosfären (t ex världshaven).
Observerade förändringar, och även prognoser av kommande förändringar, i
det globala klimatet visar på ett möjligt skifte i den nuvarande balansen
mellan kryosfären och hydrosfären, speciellt på höga latituder. Genom
kraftbalansstudier av glaciala flöden är det möjligt att upptäcka denna
obalans och på så sätt erhålla en tidig varningssignal för outtalade effekter i
områden som är känsliga för klimatförändringar (t ex Arktis och Antarktis).
Stommen i arbetet bestod av insamlandet och analys av geofysiska data i
två faser. För det första testas och evalueras modellen för
spänningsdistributioner på Storglaciären följt av termodynamiska
experiment på Bonnevie-Svendsenbreen och Kibergbreen i syfte att
undersöka vikten av spänningsinducerand uppvärmning av is med stor
tjocklek. För det andra utförs två applikationer av kraftbudgetmodellen som
59
med hjälp av testerna etablerar massbalansförhållandet mellan inflöde och
utflöde på Bonnevie-Svendsenbreen och Plogbreen i Antarktis.
Fältdata insamlades genom upprepade DGPS (Differential Global
Positioning System) och GPR (Ground Penetrating Radar) observationer på
Storglaciären från juli 2000 till september 2001 och på Kibergbreen och
Plogbreen under SWEDARP's (Swedish Antarctic Research Programme)
expedition till Drottning Maud Land 2002-2003. Dessutom tillfördes arbetet
information genom fjärranalysstudier, i sht med AVHRR-baserade digitala
höjdmodeller framtagna genom fotoklinometri och studier av ytelement med
RADARSAT-1 synthetic aperture radar (SAR), SPOT pankromatisk film
och Landsat TM.
Resultaten visar på styrkan i kraftbudgetmetodiken, främst genom
kapaciteten att urskilja både spatiella och temporala variationer i
isdynamiken given tillräcklig bakgrundsinformation. Modellen understryker
specifikt säsongsinfluenser och berggrundstopografins effekter på det
glaciala flödet.
Det påvisades dessutom att istemperaturen ökar dramatiskt under isflöden
med höga drivspänningar (>0.4 MPa) på grund av deformationsbetingad
uppvärmning. Denna effekt kan ge en positiv återkoppling, med alltjämt
tilltagande deformation, så länge den överskrider effekten av advektion av
kall is från ytan. Därigenom ger den en (del)förklaring till mekanismen
bakom snabbt flödande isströmmar i t ex Antarktis.
Massflödesberäkningarna från Bonnevie-Svendsenbreen gav ett
balanserat flöde på 0.041±0.004 km3 år-1, samt en ackumulation i
dräneringsområdet motsvarande 0.50±0.05 meter vattenekvivalent per år och
bekräftade samtidigt att det utflöde som beräknats genom kraftbudgeten kan
användas som ett mått på inflödet om systemet antas vara i jämvikt.
För Plogbreen beräknades inflödet till 0.48±0.1 km3 år-1 och utflödet till
0.55±0.05 km3 år-1. Detta ger en indikation på en negativ massbalans för
Plogbreen och kan spekulativt förklaras genom en observerad negativ
accumulationstrend (Isaksson and Karlén, 1999), men kanske främst genom
dess närhet till grundningslinjen vari den senaste tidens globala
havsyteförhöjning troligtvis inducerat en acceleration i glaciären på grund av
en minskning i de resistiva spänningaran vid den undersökta tvärsektionen.
Det sistnämnda resultatet är jämförbart med en mängd andra Antarktiska
studier som rapporterar negativa massbalanser, från t ex WAIS (West
Antarctic Ice Sheet), orsakade av globala förändringar i det atmosfäriska
cirkulationsmönstret.
60
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64
Appendix A: Symbols
Table A:1 Symbols used in the thesis.
Symbol
x, y
z
s
b
h
a
t
T
B
c
K
Q
U, u, v
w
F
L
G
V, W
V'
H
R
Wd
Wb
ø
Definition
horizontal dimensions
vertical dimension
ice surface elevation
ice bed elevation
ice depth
ice accumulation rate
time
ice temperature
resistance to ice flow, viscosity
specific heat capacity
thermal conductivity
heat production
horizontal ice velocity
vertical ice velocity
net force
lithostatic stress
Kronecker delta, a logic operator
shear and normal stress
deviatoric stress
strain rate
resistive stress
driving stress
basal resistive stress
ice flux
65
Unit
m
m
m a.s.l.
m a.s.l.
m
m w.e.a-1
yr
K
Pa a-(1/n)
J kg-1 K-1
W m-1 K-1
J
m a-1
m a-1
Pa
Pa
Pa
Pa
a-1
Pa
Pa
Pa
3 -1
m a
Appendix B: Constants
Table B:1 Constants used in this thesis.
Symbol
n
P
U
g
Qa
B0
T0
G
Hice
˜T/˜z
Value
3*
8.321
900
9.82
78.8
1.928
3155
50
3.2
0.02
Definition
flow exponent for ice
the universal gas constant
ice density
gravitational constant
activation energy for creep
empirical data
empirical data
geothermal flux
dielectric permittivity of ice
gradient at cryosphere boundary
*
Unit
J mol-1 K-1
kg m-3
m s-2
kJ mol-1
Pa a1/3
K
mW m-2
qC m-1
various values of n (given in the relevant papers), were used in respect to cases of
known conditions.
66
Appendix C: Force budget
The force balance algorithm solves for the overall spatial balance of stresses
in an isothermal block of the ice. The theory of this scheme involves a fair
amount of notations of which the normal stress and shear stress components
by V, driving stress by Wd, resistive stress by R, basal drag by Wb, and
deviatoric stresses by V’; constants such as ice density, U, gravitational
acceleration, g, flow exponent, n; and others are explained when introduced.
The flow of glaciers is driven by gravity and opposed by resistive forces
such that the net force acting on any section of a glacier is zero. This is true
when inertial effects are negligible (Van der Veen and Whillans, 1989). By
balancing forces in the directions of the coordinate axes, we obtain the
stress-equilibrium equations. Consider forces in the x-direction on an ice
block of size wx wy wz as shown in Figure C:1.
Figure C:1 Stresses on a block of size wx wy wz. (a) Normal stresses. (b) Shear
stresses (modified from Hooke, 1998).
The sum of all forces, 6Fx, acting on an ice body must equal zero according
to:
wV
§
·
¦ Fx V w y w z ¨ V xx xx w x ¸w y w z
©
wx
¹
w V yx
§
·
V w yx w x w z ¨¨ V yx w y ¸¸ w x w z
wy
©
¹
w V zx ·
§
V w zx w x w y ¨ V zx wz ¸ wxwy
wz
©
¹
U g x wxwywz 0
67
(C:1).
The normal forces are represented by the first two terms on the right. The
four following terms are shear forces in the x-direction on faces normal to
the y- and z-axes. The last term is the body force, where gx represents the
component of the gravitational acceleration parallel to the x-axis. Note that
force is obtained by multiplying stress by the area of the face in each case.
Similar force balances are set up for the y- and z-direction where, after
dividing by wx wy wz and canceling like terms of opposite sign, the balance is
kept by Equation C:2a, b, c:
wV xx wV yx wV zx
Ug x
wx
wy
wz
wV xy wV yy wV zy
Ug y
wx
wy
wz
wV xz wV yz wV zz
Ug z
wz
wy
wx
0
(C:2a, b, c).
0
0
These are the continuum equations expressing the force balance in terms of
full stresses. However, full stresses are partitioned into lithostatic (L) and
resistive stresses (Rij), where z is the elevation so that z=b at the bed and z=s
at the surface:
V ij
Rij G ij L
L
Ug (s z )
G ij 1
G ij
0
i j
iz j
(C:3a, b).
The element Rij represents resistive stress tensors and its net effect work to
oppose the movement of the glacier. The Kronecker delta, G, is a logic
operator including the lithostatic stress, L, in the case of normal stress only
(Hooke, 1998). The partitioning of full stresses into lithostatic and resistive
parts allows a clearer distinction between action (driving stress) and reaction
(gradients in resistive stresses). The action derives from gravity and the
reactions from forces applied at the margins of the studied section of an ice
body, i.e. frictional resistance to flow. By incorporating resistive and
lithostatic stresses in the system, the force balance develops into Equation
C:4a, b, c:
w>Rxx Ug ( s z )@ wRxy wRxz
0
wx
wy
wz
wRxy w R yy Ug ( s z ) wR yz
0
wx
wy
wz
wRxz wR yz w>Rzz Ug ( s z )@
0
wx
wy
wz
>
@
(C:4a, b, c).
68
Due to the softness of ice, the vertical normal stress nearly consists of
lithostatic stress alone. The resistive vertical stress, Rzz, develops as a block
of ice is influenced by the bridging effect. It has been shown that this effect
is not significant (Van der Veen and Whillans, 1989). Hence the vertical
resistive stress, Rzz, is set to zero and the balance equations can be written as
to express only the balance in horizontal forces. Integrating from the base of
a column (z=b) to the glacier surface (z=s), gives Equation C:5a, b:
s
s wR
wRxx
ws
xy
wz ³
wz Ug ( s b) Rxz ( s ) Rxz (b) 0
wx
wy
wx
b
b
³
s wR
s wR
(C:5a, b).
ws
³ wx wz ³ wy wz Ug ( s b) wx R yz ( s) R yz (b) 0
b
b
xy
yy
By reversing the order of integration and differentiation one can define basal
drag, Wbi (i=x,y), to include all basal resistance:
Riz Rxi (b)
W bi
wb
wb
Ryi (b)
wx
wy
(C:6),
where xx{x and xy{y. By substitution of the above equations into the two
integrals (Equations C:5a, b), and at the same time applying the upper
boundary condition of a stress free surface, Equations C:7a, b is obtained:
h
h
w
w
Rxx wz ³ Rxy wz W dx W bx
³
wy b
wx b
h
0
h
w
w
Rxy wz ³ R yy wz W dy W by
³
wy b
wx b
0
(C:7a, b).
These are the relationships used for calculating basal drag from driving
stress and resistive stresses acting on vertical surfaces.
69
Appendix D: Flow chart
Input data:
Force budget
Surface gradients:
Ui, s, b, h,
t, Į, T, a
˜z/˜ij, H ij, H e,
i = x, y
j = x, y, z
physical constants
Surface deviatoric stress:
ı'ij, ı'e
interpolation
matrix constr.
isothermal
glacier?
No
use lower quartile of B for
force budget calculations
Data preparation:
Driving stress:
IJdi
Surface resistive stress:
Rij
Basal resistive stress
gradients:
˜/˜i Rij
Yes
Basal drag:
Establish
temperature
profiles using
thermodynamical model
Assume
constant T
with depth
IJbi
Stress with depth:
IJxz, IJe
Viscosity with
depth
B
Thermodynamics
Ice flux with
Glen's law
Strain rate with depth:
H xz, H e
Velocity with depth and
ice flux:
Ux, Øout
Iteration to calibrate Øout
to match case specific
conditions (if known).
Note: The ice body is assigned a grid system and spatial resolution is chosen
according to the data availability and the vertical depth integration step length is
adjusted to fit observed ice depths in conjunction with desired configuration of the
thermodynamical model. The force balance considers each point/cell in the grid
system, which is why, in practice, the computations are performed matrix wise. The
computations were scripted for a PC-based OS using Mathworks© MATLAB's
capacity to perform operations on matrices (Moler, 1999).
70
Acta Universitatis Upsaliensis
Comprehensive Summaries of Uppsala Dissertations
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A doctoral dissertation from the Faculty of Science and Technology, Uppsala
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